Data encoding and decoding using Slepian-Wolf coded nested quantization to achieve Wyner-Ziv coding

ABSTRACT

A system and method for realizing a Wyner-Ziv encoder may involve the following steps: (a) apply nested quantization to input data from an information source in order to generate intermediate data; and (b) encode the intermediate data using an asymmetric Slepian-Wolf encoder in order to generate compressed output data representing the input data. Similarly, a Wyner-Ziv decoder may be realized by: (1) applying an asymmetric Slepian-Wolf decoder to compressed input data using side information to generate intermediate values, and (b) jointly decoding the intermediate values using the side information to generate decompressed output data.

PRIORITY DATA AND CONTINUATION DATA

This application is a divisional of U.S. patent application Ser. No.11/086,778, filed Mar. 22, 2005 now U.S. Pat. No. 7,295,137, entitled“Data Encoding and Decoding Using Slepian-Wolf Coded Nested Quantizationto Achieve Wyner-Ziv Coding”, invented by Liu, Cheng, Liveris and Xiong,which is a continuation-in-part of U.S. patent application Ser. No.11/068,737, filed on Mar. 1, 2005, entitled “Data Encoding and DecodingUsing Slevian-Wolf Coded Nested Quantization to Achieve Wyner-ZivCoding”, invented by Liu, Cheng, Liveris and Xiong, now U.S. Pat. No.7,256,716, and which claims the benefit of priority to U.S. ProvisionalApplication No. 60/657,520, filed on Mar. 1, 2005. Application Ser. No.11/086,778 is hereby incorporated by reference in its entirety.Application Ser. No. 11/068,737 including all its Appendices is herebyincorporated by reference in its entirety. U.S. Provisional ApplicationNo. 60/657,520, filed on Mar. 1, 2005 including all its Appendices ishereby incorporated by reference in its entirety.

STATEMENT OF U.S. GOVERNMENT LICENSING RIGHTS

The U.S. Government has a paid-up license in this invention and theright in limited circumstances to require the patent owner to licenseothers on reasonable terms as provided for by the terms of grant numberCCR-01-04834 awarded by the National Science Foundation (NSF).

FIELD OF THE INVENTION

The present invention relates to the field of informationencoding/decoding, and more particularly to a system and method forrealizing a Wyner-Ziv code using nested quantization and Slepian Wolfcoding.

DESCRIPTION OF THE RELATED ART

In 1976, Wyner and Ziv [1] established a theorem regarding the bestpossible source coding performance given distortion under the assumptionthat the decoder has access to side information. Unfortunately, codesrealizing or approaching this best possible performance have notheretofore been demonstrated. Thus, it would be greatly desirable to beable to design codes (especially practical codes) realizing orapproaching this best possible performance, and, to deploy such codesfor use in encoders and decoders.

SUMMARY

In one set of embodiments, a system and method for generating compressedoutput data may involve:

-   -   (a) receiving input data from an information source;    -   (b) applying nested quantization to the input data in order to        generate intermediate data;    -   (c) encoding the intermediate data using an asymmetric        Slepian-Wolf encoder in order to generate compressed output data        representing the input data; and    -   (d) performing at least one of storing the compressed output        data, and, transferring the compressed output data.        The values of the input data may be interpreted as vectors in an        n-dimensional space, where n is greater than or equal to one.

The information source may be a continuous source or a discrete source.A discrete source generates values in a finite set. A continuous sourcegenerates values in a continuum.

The operations (b) and (c) may be arranged so as to realize the encoderportion of a Wyner-Ziv code.

The compressed output data may be stored in a memory medium for futuredecompression. Alternatively, the compressed output data may betransferred to a decoder for more immediate decompression.

The process of applying nested quantization to the input data mayinclude: quantizing values of the input data with respect to a finelattice to determine corresponding points of the fine lattice; andcomputing indices identifying cosets of a coarse lattice in the finelattice corresponding to the fine lattice points. The intermediate datainclude said indices. The coarse lattice is a sublattice of the finelattice.

In any given dimension, some choices for the fine lattice and coarselattice may lead to better performance than others. However, theprinciples of the present invention may be practiced with non-optimalchoices for the fine lattice and coarse lattice as well as with optimalchoices.

In another set of embodiments, a system and method for recoveringinformation from compressed input data may involve:

-   -   (a) receiving compressed input data, wherein the compressed        input data is a compressed representation of a block of samples        of a first source X;    -   (b) receiving a block of samples of a second source Y;    -   (c) applying an asymmetric Slepian-Wolf decoder to the        compressed input data using the block of samples of the second        source Y, wherein said applying generates a block of        intermediate values;    -   (d) performing joint decoding on each intermediate value and a        corresponding sample of the block of second source samples to        obtain a corresponding decompressed output value.        The operations (c) and (d) may be arranged so as to realize the        decoder portion of a Wyner-Ziv code.

The joint decoding may involve determining an estimate of a centroid ofa function restricted to a region of space corresponding to theintermediate value. The function may be the conditional probabilitydensity function of the first source X given said corresponding sampleof the second source block. The centroid estimate may be (or maydetermine) the decompressed output value.

The region of space is a union of cells (e.g., Voronoi cells)corresponding to a coset of a coarse lattice in a fine lattice, whereinthe coset is identified by the intermediate value.

In yet another set of embodiments, a system and method for computing atable representing a nested quantization decoder may involve:

-   -   (a) computing a realization z of a first random vector;    -   (b) computing a realization y of a second random vector;    -   (c) adding z and y to determine a realization x of a source        vector;    -   (d) quantizing the realization x to a point in a fine lattice;    -   (e) computing an index J identifying a coset of a coarse lattice        in the fine lattice based on the fine lattice point;    -   (f) adding the realization x to a cumulative sum corresponding        to the index J and the realization y;    -   (g) incrementing a count value corresponding to the index J and        the realization y;    -   (h) repeating operations (a) through (g) a number of times;    -   (i) dividing the cumulative sums by their corresponding count        values to obtain resultant values; and    -   (j) storing the resultant values in a memory.

In one set of embodiments, a system for generating compressed outputdata may include a memory and a processor. The memory is configured tostore data and program instructions. The processor is configured to readand execute the program instructions from the memory. In response toexecution of the program instructions, the processor is operable to: (a)receive input data from an information source; (b) apply nestedquantization to the input data in order to generate intermediate data;(c) encode the intermediate data using an asymmetric Slepian-Wolfencoder in order to generate compressed output data representing theinput data; and (d) perform at least one of: storing the compressedoutput data; and transferring the compressed output data.

In another set of embodiments, a system for decoding compressed data mayinclude a memory and processor. The memory is configured to store dataand program instructions. The processor is configured to read andexecute the program instructions from the memory. In response toexecution of the program instructions, the processor is operable to: (a)receive compressed input data, wherein the compressed input data is acompressed representation of a block of samples of a first source X; (b)receive a block of samples of a second source Y; (c) apply an asymmetricSlepian-Wolf decoder to the compressed input data using the block ofsamples of the second source Y, wherein said applying generates a blockof intermediate values; (d) perform joint decoding on each intermediatevalue and a corresponding sample of the block of second source samplesto obtain a corresponding decompressed output value, wherein saidperforming joint decoding includes determining an estimate of a centroidof a function restricted to a region of space corresponding to theintermediate value, wherein said estimate determines the decompressedoutput value. The function is the conditional probability densityfunction of the first source X given said corresponding sample of thesecond source block.

In yet another set of embodiments, a system for computing a tablerepresenting a nested quantization decoder may include a memory andprocessor. The memory is configured to store data and programinstructions. The processor is configured to read and execute theprogram instructions from the memory. In response to execution of theprogram instructions, the processor is operable to: (a) computing arealization z of a first random vector; (b) computing a realization y ofa second random vector; (c) adding z and y to determine a realization xof a source vector; (d) quantizing the realization x to a point in afine lattice; (e) computing an index J identifying a coset of a coarselattice in the fine lattice based on the fine lattice point; (f) addingthe realization x to a cumulative sum corresponding to the index J andthe realization y; (g) incrementing a count value corresponding to theindex J and the realization y; (h) repeating operations (a) through (g)a number of times; (i) dividing the cumulative sums by theircorresponding count values to obtain resultant values; and (j) storingthe resultant values in a memory medium.

We propose a practical scheme that we refer to as Slepian-Wolf codednested quantization (SWC-NQ) for Wyner-Ziv coding that deals with sourcecoding with side information under a fidelity criterion. The schemeutilizes nested lattice quantization with a fine lattice forquantization and a coarse lattice for channel coding. In addition, atlow dimensions (or block sizes), an additional Slepian-Wolf coding stageis added to compensate for the weakness of the coarse lattice channelcode. The role of Slepian-Wolf coding in SWC-NQ is to exploit thecorrelation between the quantized source and the side information forfurther compression and to make the overall channel code stronger.

The applications of this proposed scheme are very broad; it can be usedin any application that involves lossy compression (e.g., of speechdata, audio data, image data, video data, graphic data, or, anycombination thereof).

We show that SWC-NQ achieves the same performance of classicentropy-constrained lattice quantization. For example, 1-D/2-D SWC-NQperforms 1.53/1.36 dB away from the Wyner-Ziv rate distortion (R-D)function of the quadratic Gaussian source at high rate assuming idealSlepian-Wolf coding. In other words, the scheme may be optimal in termsof compression performance, at least in some embodiments. We alsodemonstrate means of achieving efficient Slepian-Wolf compression viamulti-level LDPC codes.

BRIEF DESCRIPTION OF THE DRAWINGS

A better understanding of the present invention can be obtained when thefollowing detailed description of the preferred embodiment is consideredin conjunction with the following drawings, in which:

FIG. 1A illustrates one embodiment of a computer system that may be usedfor implementing various of the method embodiments described herein;

FIG. 1B illustrates one embodiment of a communication system includingtwo computers coupled through a computer network;

FIG. 2 is a block diagram for one embodiment of a computer system thatmay be used for implementing various of the method embodiments describedherein;

FIG. 3A illustrates one embodiment of a sensor system as a possibleapplication of the inventive principles described herein;

FIG. 3B illustrates one embodiment of a video transmission as anotherpossible application of the inventive principles described herein;

FIG. 3C illustrates a system that compressed source information andstored the compressed information in a memory medium for later retrievaland decompression;

FIG. 4 illustrates one embodiment of a method for encoding data;

FIG. 5 illustrates one embodiment of a method for decoding data usingside information;

FIG. 6 illustrates one embodiment of a method for computing a table thatrepresents an nested quantization decoder.

FIG. 7 illustrates an example of a fine lattice, coarse lattice, cosetleader vector v and region R(v) in dimension n=2;

FIG. 8 illustrates a simplified nested quantization ender and decoder;

FIG. 9 shows δ₂(R) with different V₂'s using nested A₂ lattices (i.e.,hexagonal lattices) in dimension n=2;

FIG. 10 shows D ₂(R) as the convex hull of δ₂(R) with different V₂;

FIG. 11 shows the granular and boundary components of distortion withdifferent V₂'s;

FIG. 12 plots D _(n)(R) for n=1, 2, 4, 8 and 24 with σ_(Z) ²=0.01;

FIG. 13 shows the lower bound of D(R) with different V₂'s in the 1-Dcase;

FIGS. 14( a) and (b) plot the optimal V₂* (scaled by σ_(Z)) as afunction of R for the 1-D (n=1) and 2-D (n=2) cases;

FIG. 15 shows the improvement gained by using the optimal (non-linear)estimator at low rates, for n=2 and σ_(Z) ²=0.01;

FIG. 16 illustrates one embodiment of a multi-layer Slepian Wolf codingscheme;

FIG. 17 shows results based on 1-D nested lattice quantization both withand without Slepian Wolf coding (SWC); and

FIG. 18 shows results based on 2-D nested lattice quantization both withand without Slepian Wolf coding (SWC).

While the invention is susceptible to various modifications andalternative forms, specific embodiments thereof are shown by way ofexample in the drawings and are herein described in detail. It should beunderstood, however, that the drawings and detailed description theretoare not intended to limit the invention to the particular formdisclosed, but on the contrary, the intention is to cover allmodifications, equivalents and alternatives falling within the spiritand scope of the present invention as defined by the appended claims.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS Incorporation byReference

The following patent application documents are hereby incorporated byreference in their entirety as though fully and completely set forthherein:

U.S. Provisional Application Ser. No. 60/657,520, titled “Multi-SourceData Encoding, Transmission and Decoding”, filed Mar. 1, 2005, whoseinventors are Vladimir M. Stankovic, Angelos D. Liveris, Zixiang Xiong,Costas N. Georghiades, Zhixin Liu, Samuel S. Cheng, and Qian Xu,including Appendices A through H;

U.S. patent application Ser. No. 11/069,935, titled “Multi-Source DataEncoding, Transmission and Decoding Using Slepian-Wolf Codes Based OnChannel Code Partitioning”, filed Mar. 1, 2005, whose inventors areVladimir M. Stankovic, Angelos D. Liveris, Zixiang Xiong, and Costas N.Georghiades, including Appendices A through H; and

U.S. patent application Ser. No. 11/068,737, titled “Data Encoding andDecoding Using Slepian-Wolf Coded Nested Quantization to AchieveWyner-Ziv Coding”, filed Mar. 1, 2005, whose inventors are Zhixin Liu,Samuel S. Cheng, Angelos D. Liveris, and Zixiang Xiong, includingAppendices A through H.

The following publications are referred to herein and are incorporatedby reference in their entirety as though fully and completely set forthherein:

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Terminology.

The following is a glossary of terms used in the present application:

-   -   Memory Medium—Any of various types of memory devices, storage        devices, or combinations thereof. The term “memory medium” is        intended to include: CD-ROM, any of various kinds of magnetic        disk (such as floppy disk or hard disk), any of various kinds of        magnetic tape, optical storage, and bubble memory; any of        various kinds of read only memory (ROM); any of various kinds of        random access memory (RAM) such as DRAM, DDR RAM, SRAM, EDO RAM,        Rambus RAM, etc.    -   Carrier Medium—a memory medium as described above, or, a        communication medium on which signals are conveyed, e.g.,        signals such as electrical, electromagnetic, acoustic, optical        signals.    -   Programmable Hardware Element—includes various types of        programmable hardware, reconfigurable hardware, programmable        logic, or field-programmable devices (FPDs), such as one or more        FPGAs (Field Programmable Gate Arrays), or one or more PLDs        (Programmable Logic Devices), or other types of programmable        hardware. A programmable hardware element may also be referred        to as “reconfigurable logic”.    -   Program—the term “program” is intended to have the full breadth        of its ordinary meaning. The term “program” includes 1) a        software program which may be stored in a memory and is        executable by a processor or 2) a hardware configuration program        useable for configuring a programmable hardware element.    -   Software Program—the term “software program” is intended to have        the full breadth of its ordinary meaning, and includes any type        of program instructions, code, script and/or data, or        combinations thereof, that may be stored in a memory medium and        executed by a processor. Exemplary software programs include        programs written in text-based programming languages, such as C,        C++, Pascal, Fortran, Cobol, Java, assembly language, etc.;        graphical programs (programs written in graphical programming        languages); assembly language programs; programs that have been        compiled to machine language; scripts; and other types of        executable software. A software program may comprise two or more        components that interoperate.    -   Hardware Configuration Program—a program, e.g., a netlist or bit        file, that can be used to program or configure a programmable        hardware element.    -   Computer System—any of various types of computing or processing        systems, including a personal computer system (PC), mainframe        computer system, workstation, network appliance, Internet        appliance, personal digital assistant (PDA), television system,        grid computing system, or other device or combinations of        devices. In general, the term “computer system” can be broadly        defined to encompass any device (or combination of devices)        having at least one processor that executes instructions from a        memory medium.

FIG. 1A—Computer System.

FIG. 1A illustrates a computer system 82, according to one set ofembodiments, operable to execute a set of programs. The programs may beconfigured to implement any or all of the method embodiments describedherein. The computer system 82 may include one or more processors,memory media, and one or more interface devices. The computer system 82may also include input and output devices. The memory media may includevarious well known systems and devices configured for the storage ofdata and computer programs. For example, the memory media may store oneor more programs which are executable to perform the methods (or somesubset of the methods) described herein. The memory medium may alsostore operating system software, as well as other software for operationof the computer system. In various embodiments, the computer system 82may be a personal computer, a notebook computer, a workstation, aserver, a router, a computer implemented on a card, etc.

FIG. 1B—Computer Network.

FIG. 1B illustrates a communication system including a first computersystem 82 and a second computer system 90, according to one set ofembodiments. The first computer system 82 couples to the second computersystem 90 through a network 84 (or, more generally, any of various knowncommunication mechanisms). The first and second computer systems mayeach be any of various types, as desired. The network 84 can also be anyof various types, including a LAN (local area network), WAN (wide areanetwork), the Internet, or an Intranet.

Each of the computer systems may be configured with programsimplementing any or all of the method embodiments described herein. Inone embodiment, the first and second computer systems are eachconfigured with software for encoding and decoding data as describedvariously herein.

It is noted that computer system 82 and computer system 90 may beconfigured according to any of various system architectures.

FIG. 2—Computer System Block Diagram.

FIG. 2 is a block diagram representing one embodiment of computer system82 and/or computer system 90.

The computer system may include at least one central processing unit CPU160 which is coupled to a host bus 162. The CPU 160 may be any ofvarious types, including, but not limited to, an x86 processor, aPowerPC processor, a CPU from the SPARC family of RISC processors, aswell as others. A memory medium, typically comprising RAM, and referredto as main memory 166, is coupled to the host bus 162 by means of memorycontroller 164. The main memory 166 may store programs operable toimplement encoding and/or decoding according to any (or all) of thevarious embodiments described herein. The main memory may also storeoperating system software, as well as other software for operation ofthe computer system.

The host bus 162 couples to an expansion or input/output bus 170 througha bus controller 168 or bus bridge logic. The expansion bus 170 may bethe PCI (Peripheral Component Interconnect) expansion bus, althoughother bus types can be used. The expansion bus 170 includes slots forvarious devices such as a video card 180, a hard drive 182, a CD-ROMdrive (not shown) and a network interface 122. The network interface 122(e.g., an Ethernet card) may be used to communicate with other computersthrough the network 84.

In one embodiment, a device 190 may also be connected to the computer.The device 190 may include an embedded processor and memory. The device190 may also or instead comprise a programmable hardware element (suchas an FPGA). The computer system may be operable to transfer a programto the device 190 for execution of the program on the device 190. Theprogram may be configured to implement any or all of the encoding ordecoding method embodiments described herein.

In some embodiments, the computer system 82 may include input devicessuch as a mouse and keyboard and output devices such a display andspeakers.

FIGS. 3A, 3B & 3C—Exemplary Systems.

Various embodiments of the present invention may be directed to sensorsystems, wireless or wired transmission systems, or, any other type ofinformation processing or distribution system utilizing the codingprinciples described herein.

For example, as FIG. 3A shows, a sensor system may include a firstsensor (or set of sensors) and a second sensor (or set of sensors). Thefirst sensor may provide signals to a transmitter 306. The sensors maybe configured to sense any desired physical quantity or set of physicalquantities such as time, temperature, energy, velocity, flow rate,displacement, length, mass, voltage, electrical current, charge,pressure, etc. The transmitter 306 may receive the signals, digitize thesignals, encode the signals according the inventive principles describedherein, and transmit the resulting compressed data to a receiver 308using any of various known communication mechanism (e.g., a computernetwork). The receiver 308 receives the compressed data from thetransmitter as well as side information from a second sensor. Thereceiver 308 decodes the compressed data, according to the inventiveprinciples described herein, using the side information, and thereby,generates decompressed output data. The decompressed output data may beused as desired, e.g., displayed to a user, forwarded for analysisand/or storage, etc.

As another example, a first video source may generate video signals asshown in FIG. 3B. A transmitter 316 receives the video signal, encodesthe video signals according the inventive principles described herein,and transmits the resulting compressed data to a receiver 318 using anyof various known communication mechanism (e.g., a computer network). Thereceiver 318 receives the compressed data from the transmitter as wellas side information from a second sensor. The receiver 318 decoders thecompressed data, according to the inventive principles described herein,using the side information.

As yet another embodiment, a encoder 326 may receive signals from afirst source and encode the source signals according to the inventiveprinciples described herein, and store the resulting compressed dataonto a memory medium 327. At some later time, an encoder 328 may readthe compressed data from the memory medium 327 and decode the compresseddata according to the inventive principles described herein.

It is noted that embodiments of the present invention can be used for aplethora of applications and is not limited to the above applications.In other words, applications discussed in the present description areexemplary only, and the present invention may be used in any of varioustypes of systems. Thus, the system and method of the present inventionis operable to be used in any of various types of applications,including audio applications, video applications, multimediaapplications, any application where physical measurements are gathered,etc.

FIG. 4 illustrates one embodiment of a method for decoding data. In step405, input data is received from an information source.

In step 410, nested quantization as described herein is applied to theinput data in order to generate intermediate data.

In step 420, the intermediate data is encoded using an asymmetricSlepian-Wolf encoder as described herein, in order to generatecompressed output data representing the input data.

The nested quantization and asymmetric Slepian-Wolf encoder may beconfigured so that the combination of steps 410 and 420 realizes theencoder portion of a Wyner-Ziv code.

In step 425, the compressed output data may be stored and/ortransferred. In one embodiment, the compressed output data may be storedonto a memory medium for decompression at some time in the future. Inanother embodiment, the compressed output data may be transferred, e.g.,to a decoder device.

The information source may be a continuous source or a discrete source.A discrete source generates values in a finite set. A continuous sourcegenerates values in a continuum. The values of the input data may beinterpreted as vectors in an n-dimensional space, where n is greaterthan or equal to one.

The process of applying nested quantization to the input data mayinclude: quantizing values of the input data with respect to a finelattice to determine corresponding points of the fine lattice; andcomputing indices identifying cosets of a coarse lattice in the finelattice corresponding to the fine lattice points. The intermediate datainclude said indices. The coarse lattice is a sublattice of the finelattice.

In any given dimension, some choices for the fine lattice and coarselattice may lead to better performance than others. However, theprinciples of the present invention may be practiced with non-optimalchoices for the fine lattice and coarse lattice as well as with optimalchoices.

In various embodiments, the information source may be a source of audioinformation, a source of video information, a source of imageinformation, a source of text information, a source of informationderived from physical measurements (e.g., by a set of one or morephysical sensors), or, any combination thereof.

As discussed in reference [29], one way to do asymmetric Slepian-Wolfencoding is by means of syndrome forming, which involves a modificationof classical channel encoding. This type of Slepian-Wolf encoding isused to generate the simulation results described in this paper.However, the general method of Slepian-Wolf coded nested quantizationdisclosed in this paper can also be performed with other forms ofSlepian-Wolf encoders.

In some embodiments, the asymmetric Slepian-Wolf encoder may be a lowdensity parity check syndrome former or a turbo syndrome former.

In one embodiment, the asymmetric Slepian-Wolf encoder may be configuredas a multi-layered encoder as described herein.

An encoder system may be configured to implement any embodiment of themethod illustrated and described above in connection with FIG. 4. Theencoder system may include one or more processors or programmablehardware elements, and/or, dedicated circuitry such as applicationspecific integrated circuits. In one embodiment, the encoder systemincludes a processor (e.g., a microprocessor) and memory. The memory isconfigured to store program instructions and data. The processor isconfigured to read and execute the program instructions from the memoryto implement any embodiment of the method illustrated and describedabove in connection with FIG. 4.

Furthermore, a computer-readable memory medium may be configured tostore program instructions which are executable by one or moreprocessors to implement any embodiment of the method illustrated anddescribed above in connection with FIG. 4.

FIG. 5 illustrates one embodiment of a method for decoding data. In step510, compressed input data is received. The compressed input data is acompression representation of a block of samples of a first source X. Instep 512, a block of samples of a second source Y is received. Steps 510and 512 need not be performed in any particular order. In oneembodiment, steps 510 and 512 may be performed in parallel, or, at leastin a time overlapping fashion. The first source X and the second sourceY may be statistically correlated.

In step 514, an asymmetric Slepian-Wolf decoder as described herein isapplied to the compressed input data using the block of samples of thesecond source Y. This application of the asymmetric Slepian-Wolf decodergenerates a block of intermediate values.

In step 516, joint decoding is performed on each intermediate value anda corresponding sample of the block of second source samples to obtain acorresponding decompressed output value. The joint decoding may includedetermining an estimate of a centroid of a function restricted to aregion of space corresponding to the intermediate value. The functionmay be the conditional probability density function of the first sourceX given said corresponding sample of the second source block. Thecentroid estimate may be (or may determine) the decompressed outputvalue. The resulting block of decompressed output values may be used inany of various ways as desired. For example, the block of decompressedoutput values may be displayed to a user, forwarded for analysis and/orstorage, transmitted through a network to one or more otherdestinations, etc.

The steps 514 and 516 may be configured so as to realize the decoderportion of a Wyner-Ziv code.

The region of space is a union of cells (e.g., Voronoi cells)corresponding to a coset of a coarse lattice in a fine lattice, whereinthe coset is identified by the intermediate value.

The centroid estimate may be determined by reading the centroid estimatefrom a table stored in a memory medium using said corresponding sampleof the second source block and the intermediate value as addresses. Thetable may be computed in at a central code design facility, and, thendeployed to a decoder system through any of various known means for datadistribution. The table may be stored in a memory medium of the decodersystem. The decoder system may accessing the table to determine thecentroid estimate in real time.

In one alternative embodiment, the centroid estimate may be determinedby performing a Monte Carlo iterative simulation at decode time.

The intermediate values generated in step 514 may specify cosets of acoarse lattice in a fine lattice. The coarse lattice may be a sublatticeof the fine lattice.

The asymmetric Slepian-Wolf decoder may be a multi-layered decoder.Furthermore, the asymmetric Slepian-Wolf decoder may be a low densityparity check decoder or a turbo decoder.

A decoder system may be configured to implement any embodiment of themethod illustrated and described above in connection with FIG. 5. Thedecoder system may include one or more processors or programmablehardware elements, and/or, dedicated circuitry such as applicationspecific integrated circuits. In one embodiment, the decoder systemincludes a processor (e.g., a microprocessor) and memory. The memory isconfigured to store program instructions and data. The processor isconfigured to read and execute the program instructions from the memoryto implement any embodiment of the method illustrated and describedabove in connection with FIG. 5.

Furthermore, a computer-readable memory medium may be configured tostore program instructions which are executable by one or moreprocessors to implement any embodiment of the method illustrated anddescribed above in connection with FIG. 5.

FIG. 6 illustrates one embodiment of a method for computing a tablerepresenting a nested quantization decoder by Monte Carlo simulation.The method may be implemented by executing program instructions on acomputer system (or a set of interconnected computer systems). Theprogram instructions may be stored on any of various knowncomputer-readable memory media.

In step 610, the computer system may compute a realization z of a firstrandom vector (the auxiliary vector), e.g., using one or more randomnumber generators. In step 615, the computer system may compute arealization y of a second random vector (the side information), e.g.,using one or more random number generators. Steps 610 and 615 need notbe performed in any particular order.

In step 620, the computer system may add the realization y and therealization z to determine a realization x of a source vector.

In step 625, the computer system may quantize the realization x to apoint p in a fine lattice as described herein.

In step 630, the computer system may compute an index J identifying acoset of a coarse lattice in the fine lattice based on the fine latticepoint p. The coarse lattice is a sublattice of the fine lattice.

The computer system may maintain a set of cumulative sums, i.e., onecumulative sum for each possible pair in the Cartesian product (CPR) ofthe set of possible indices and the set of possible realizations of thesecond random vector (the side information). The cumulative sums may beinitialized to zero. Furthermore, the computer system may maintain a setof count values, i.e., one count value for each possible pair in theCartesian product CPR.

In step 635, the computer system may add the realization x to acumulative sum corresponding to the index J and the realization y. Instep 640, the computer system may increment a count value correspondingto the index J and the realization y. Steps 635 and 640 need not beperformed in any particular order.

The computer system may repeat steps 610 through 640 a number of timesas indicated in step 645. In one embodiment, the number of repetitionsmay be determined by input provided by a user.

In step 650, the computer system may divide the cumulative sums by theircorresponding count values to obtain resultant values. The resultantvalues may be interpreted as being the centroid estimates describedabove in connection with FIG. 5.

In step 655, the computer system may store the resultant values as atable in a memory associated with the computer system, e.g., onto harddisk.

The table may be distributed (e.g., with decoding software configuredaccording to any of the various method embodiments described herein) todecoder systems by any of various means. In one embodiment, the tablemay be downloaded to decoder systems over a network such as theInternet. In another embodiment, the table may be stored on acomputer-readable memory media (such as CD-ROM, magnetic disk, magnetictape, compact flash cards, etc.) and the memory media may be provided(e.g., sold) to users of decoder systems for loading onto theirrespective computer systems.

In one embodiment, system for computing a table representing a nestedquantization decoder may be configured with a processor and memory. Thememory is configured to store program instructions and data. Theprocessor is configured to read and execute the program instructionsfrom the memory to implement any embodiment of the method illustratedand described above in connection with FIG. 6.

The Wyner-Ziv coding problem deals with source coding with sideinformation under a fidelity criterion. The rate-distortion function forthis setup, R*(D), is given by [1]:

$\begin{matrix}{{{R^{*}(D)} = {\min\limits_{p{({u❘x})}}{\min\limits_{{f\text{:}A_{U} \times A_{Y}}->A_{\hat{X}}}\left\lbrack {{I\left( {U;X} \right)} - {I\left( {U;Y} \right)}} \right\rbrack}}},} & (1)\end{matrix}$where the source X (with an alphabet A_(X)), the side information Y(with an alphabet A_(Y)) and the auxiliary random variable U (with analphabet Au) form a Markov chain as Y

X

U, with the distortion constraint E[d(X,ƒ(U,Y),Y)]≦D. The function I(·)denotes the Shannon mutual information as defined in [3]. The functionp(u|x) is the conditional probability of U given X. The function frepresents the mapping from the possible auxiliary variable and sideinformation to a reconstructed value of X.

Although the theoretical limits for the rate-distortion function havebeen known for some time [1], [2], practical approaches to binaryWyner-Ziv coding and continuous Wyner-Ziv coding have not appeared untilrecently [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. A commoncontext of interest for continuous Wyner-Ziv coding is code design forthe quadratic Gaussian case, where the correlation between the source Xand the side information Y is modeled as an additive white Gaussiannoise (AWGN) channel as X=Y+Z, Z→N(0, σ_(Z) ²), with a mean-squarederror (MSE) measure. For this case, one can first consider lattice codes[13] or trellis-based codes [14], [15] that have been used for bothsource and channel coding in the past, and focus on finding good nestingcodes among them. Following Zamir et al.'s nested lattice coding scheme[16], Servetto [3] proposed explicit nested lattice constructions basedon similar sublattices [17] with the assumption of high correlation.Research on trellis-based nested codes as a way of realizinghigh-dimensional nested lattice codes has just started recently [7]. Forexample, in DISCUS [7], two source codes (scalar quantization andtrellis coded quantization—TCQ) and two channel codes (scalar coset codeand trellis-based coset code [14]) are used in source-channel coding forthe Wyner-Ziv problem, resulting in four combinations. One of them(scalar quantization with scalar coset code) is nested scalarquantization and another one (TCQ with trellis-based coset code, alsosuggested in [4]) can effectively be considered as nested TCQ.

Zamir et al. [18], [16] first outlined some theoretical constructionsusing a pair of nested linear/lattice codes for binary/Gaussian sources,where the fine code in the nested pair plays the role of source codingwhile the coarse code does channel coding. They also proved that, forthe quadratic Gaussian case, the Wyner-Ziv rate-distortion (R-D)function is asymptotically achievable using nested lattice codes, withthe assumption that the lattice is ideally sphere-packed as the latticedimension goes to infinity.

The performance of a nested lattice quantizer can approach the Wyner-Zivlimit at high rate when high-dimensional lattices are used, because boththe granular gain and boundary gain reach their ultimate values [19]when the dimension n→∞. Nevertheless, lattice coding and code designwith high dimensionality are difficult in practice.

For a nested lattice quantizer using low-to-moderate dimensionallattices, a pragmatic approach to boost the overall performance is toincrease the boundary gain with a second stage of binning, withoutincreasing the dimensionality of the lattices. Suppose a second stage ofbinning is applied without introducing extra overload probabilityP_(ol), and the binning scheme partitions the support region of finelattice (actually the Voronoi region of the coarse lattice for nestedlattice quantizer) into m cosets. Thus the volume of the support regiondecreases by a factor of N/m while the overload probability stays fixed,where N is the nesting ratio. From the definition of boundary gain [19],the boundary gain increases without changing the dimension of thelattices. Since various possible boundary gains are realizable using thesecond-stage of binning as discussed above, there is only maximally 1.53dB (decibels) of granular gain left unexploited by the quantizer. Thusthe second stage of binning allows us to show the theoreticalperformance limits at high rates with low-to-moderate dimensional sourcecodes.

In this paper, we introduce a new framework for the continuous Wyner-Zivcoding of independent and identically distributed (i.i.d.) sources baseda combination of Slepian-Wolf coding (SWC) and nested quantization (NQ).In this framework, which we refer to as SWC-NQ, the role of Slepian-Wolfcoding, as a second-stage of binning which increases the boundary gainof source coding, is to exploit the correlation between the quantizedsource and the side information for further compression and by makingthe overall channel code stronger.

SWC-NQ connects network information theory with the rich areas of (a)lattice source code designs (e.g., [13]) and (b) channel code designs(e.g., LDPC codes [20], [21] and [22]), making it feasible to devisecodes that can approach the Wyner-Ziv rate-distortion function. LDPC isan acronym for “low density parity check”.

For the quadratic continuous case, we establish the high-rateperformance of SWC-NQ with low-to-moderate dimensional nestedquantization and ideal SWC. We show that SWC-NQ achieves the sameperformance of classic entropy-constrained lattice quantization as ifthe side information were also available at the encoder. For example,1-D/2-D SWC-NQ performs 1.53/1.36 dB away from the Wyner-Ziv R-Dfunction of the quadratic continuous source at high rate assuming idealSWC.

A recent work, [23], starts with non-uniform quantization with indexreuse and Slepian-Wolf coding and shows the same high-rate theoreticalperformance as ours when the quantizer becomes an almost uniform onewithout index reuse. This agrees with our assertion that at high rates,the nested quantizer asymptotically becomes a non-nested regular one sothat strong channel coding is guaranteed.

We also implement 1-D and 2-D nested lattice quantizers in the raterange of 1-7 bits per sample. Although our analysis shows that nestingdoes not help at high rate, experiments using nested lattice quantizerstogether with irregular LDPC codes for SWC obtain performances close tothe corresponding limits at low rates. Our work thus shows that SWC-NQprovides an efficient scheme for practical Wyner-Ziv coding withlow-dimensional lattice quantizers at low rates.

Although the theoretical analyses are taken under the assumption of highrate, the rate-distortion performance at low rate is still consistentwith the one at high rate, i.e., SWC-NQ achieves the same performance ofclassic entropy coded quantization (ECQ) as if the side information werealso available at the encoder even at low rate, when a non-linearestimator is applied at the decoder. This non-linear estimator, as wepresent in this paper, is the optimal one in the sense of the MSEmeasurement. At high rates, the non-linear estimator reduces to thelinear one analyzed in this paper.

We note that the non-linear estimation in the decoder can yieldsignificant gains for low rates and for high rates it cannot helpnoticeably. This is confirmed by the agreement of the high rate analysisresults in this paper, which assume that the linear estimation is used,with the high rate simulation results, for which the non-linearestimation method is always used.

The following is a list of some of the contents of this paper:

1. A theoretical analysis and simulation for low-to-moderate dimensionalnested lattice quantization at high rates. The rate-distortion functionfor general continuous sources with arbitrary probability densityfunction (PDF) and MSE measurement, and a theoretical lower bound ofrate-distortion function for the quadratic Gaussian case, are presented.

2. An analysis of the granular and boundary gains of the source codingcomponent of nested lattice quantization. This analysis explains thephenomenon of an increasing gap of the rate-distortion function ofnested lattice quantization at low-to-moderate dimension, with respectto the Wyner-Ziv limit, as we observe in the simulation.

3. A new Wyner-Ziv coding framework using nested lattice quantizationand Slepian-Wolf coding, which we refer to as SWC-NQ, is introduced. TheSWC-NQ rate-distortion function for general continuous sources witharbitrary PDF and MSE measurement is presented, and is in agreement withthe performance of entropy-constrained lattice quantization as if theside information were available at the encoder.

4. A non-linear estimator for the decoder corresponding to the nestedquantizer is presented, and is proved to be optimal in sense of MSEmeasurement. This estimator helps to improve the performance of SWC-NQat low rates, and is consistent with the analytical performance at highrates.

5. Examples of practical code design using a 1-D (scalar) lattice and2-D (hexagonal) lattice, and multi-layer irregular LDPC codes, are givenin this paper.

Some Background on Wyner-Ziv Coding.

In this section, we briefly review the basic concepts and milestonetheorems of Wyner-Ziv coding. Wyner and Ziv [1], [2] present the limitof rate-distortion performance for lossy coding with side information,for both Gaussian and general sources.

The problem of rate distortion with side information at the decoder asksthe question of how many bits are needed to encode X under theconstraint that E[d(X,{circumflex over (X)})]≦D, assuming the sideinformation Y is available at the decoder but not at the encoder. Thisproblem generalizes the setup of [24] in that coding of X is lossy withrespect to a fidelity criterion rather than lossless. For both discreteand continuous alphabets of A_(X) and general distortion metrics d(·),Wyner and Ziv [1] gave the rate-distortion function R_(WZ)*(D) for thisproblem as R_(WZ)*(D)=inf I(X; Z|Y), where the infimum is taken over allrandom variables Z such that Y→X→Z is a Markov chain and there exists afunction {circumflex over (X)}=X(Z,Y) satisfying E[d(X,{circumflex over(X)})]≦D. According to [1],

${{R_{WZ}^{*}(D)} \geq {R_{X❘Y}(D)}} = {\inf\limits_{\{{\hat{X} \in {{A_{X}\text{:}{E{\lbrack{d{({X,\hat{X}})}}\rbrack}}} \leq D}}\}}{{I\left( {X;{\hat{X}❘Y}} \right)}.}}$

This means that usually there is a rate loss in the Wyner-Ziv problem.Zamir quantified this loss in [25]. In particular, Zamir showed a rateloss of less than 0.22 bit for a binary source with Hamming distance,and a rate loss of less than 0.5 bit/sample for continuous sources withMSE distortion.

Note that when D=0, the Wyner-Ziv problem degenerates to theSlepian-Wolf problem with R_(WZ)*(0)=R_(X|Y)(0)=H(X|Y). Another specialcase of the Wyner-Ziv problem is the quadratic Gaussian case when X andY are zero mean and stationary Gaussian memoryless sources and thedistortion metric is MSE. Let X_(i) denote the i^(th) component of X,and Y_(i) denotes the i^(th) component of Y, i=1, 2, . . . , n. Let thecovariance matrix of (X_(i), Y_(i)) be

${{cov}\left( {X_{i},Y_{i}} \right)} = \begin{bmatrix}\sigma_{X}^{2} & {\rho\;\sigma_{X}\sigma_{Y}} \\{\rho\;\sigma_{X}\sigma_{Y}} & \sigma_{Y}^{2}\end{bmatrix}$with |ρ|<1 for all n, then

${{R_{WZ}^{*}(D)} = {{R_{X❘Y}(D)} = {\frac{1}{2}{\log^{+}\left\lbrack \frac{\sigma_{X}^{2}\left( {1 - \rho^{2}} \right)}{D} \right\rbrack}}}},$where log⁺ x=max {0, log x}. This case is of special interest inpractice because many image and video sources can be modeled as jointlyGaussian (after mean subtraction) and Wyner-Ziv coding suffers no rateloss.

Lattices and Nested Lattices.

In this section, we review the idea of lattice and nested lattices andintroduce notation that will be used in our discussion.

For a set of n basis vectors {g₁, . . . , g_(n),} in R^(n), an unboundedn-dimensional (n-D) lattice A is defined byΛ={l=Gi:i∈Z^(n)}  (2)and its generator matrixG=[g₁|g₂| . . . |g_(n)].R denotes the set of real numbers. R^(n) denotes n-dimensional Euclideanspace. Z denotes the set of integers. Z^(n) denotes the Cartesianproduct of n copies of Z, i.e., the set of n-vectors whose componentsare integers.

The nearest neighbor quantizer Q_(Λ)(x) associated with Λ is given by

$\begin{matrix}{{Q_{\Lambda}(x)} = {\arg{\min\limits_{l \in \Lambda}{{{x - l}}.}}}} & (3)\end{matrix}$

The notation “arg min” denotes the value of the argument (in this casel) where the minimum is achieved. Expression (3) is augmented with a setof “tie breaking” rules to decide the result in cases where two or morepoints of the lattice Λ achieve the minimum distance to vector x. Any ofvarious sets of tie breaking rules may be used. For example, indimension one (i.e., n=1) with lattice Λ being the integers, points ofthe form k+(½) with be equidistant to k and k+1. One possibletie-breaking rule would be to map such points up to k+1. In one set ofembodiments, the nearest neighbor quantizer defined by (3) and a set oftie breaking rules has the property:Q _(Λ)(x+l)=Q _(Λ)(x)+l, ∀l∈Λ.

The basic Voronoi cell of Λ, which specifies the shape of thenearest-neighbor decoding region, isK={x:Q _(Λ)(x)=0}.  (4)

Associated with the Voronoi cell K are several important quantities: thecell volume V, the second moment σ² and the normalized second momentG(Λ), defined by

$\begin{matrix}{{V = {\int_{K}\ {\mathbb{d}x}}},} & (5) \\{{\sigma^{2} = {\frac{1}{nV}{\int_{K}{{x}^{2}\ {\mathbb{d}x}}}}},} & (6) \\{{{G(\Lambda)} = \frac{\sigma^{2}}{V^{2/n}}},} & (7)\end{matrix}$respectively. The minimum of G(Λ) over all lattices in R^(n) is denotedas G_(n). By [13],

$\begin{matrix}{{G_{n} \geq \frac{1}{\left( {2{\pi\mathbb{e}}} \right)}},{\forall n}} & (8) \\{{\underset{n\rightarrow\infty}{\lim\mspace{11mu}}G_{n}} = \frac{1}{\left( {2{\pi\mathbb{e}}} \right)\;}} & (9)\end{matrix}$

The notation “∀” is to be read as “for all”. The constant e is Euler'sconstant.

A pair of n-D lattices (Λ₁,Λ₂) with corresponding generator matrices G₁and G₂ is nested, if there exists an n×n integer matrix P such thatG ₂ =G ₁ ×P anddet{P}>1,where det{P} denotes the determinant of the matrix P. In this case V₂/V₁is called the nesting ratio, and Λ₁ and Λ₂ are called the fine latticeand coarse lattice, respectively.

For a pair (Λ₁,Λ₂) of nested lattices, the points in the setΛ₁/Λ₂≡{Λ₁∩K₂} are called the coset leaders of Λ₂ relative to Λ₁, whereK₂ is the basic Voronoi cell of Λ₂. The notation “A≡B” means that A isbeing defined by expression B, or vice versa. For each v∈Λ₁/Λ₂ the setof shifted lattice pointsC(v)≡{v+l, ∀l∈Λ ₁}is called a coset of Λ₂ relative to Λ₁. The j^(th) point of C(v) isdenoted as c_(j)(v). Then

$\begin{matrix}{{{C(0)} = {\left\{ {{c_{j}(0)},{\forall{j \in Z}}} \right\} = \Lambda_{2}}},\;{and}} & (10) \\{{\underset{v \in {\Lambda_{1}/\Lambda_{2}}}{Y}{C(v)}} = {\Lambda_{1}.}} & (11)\end{matrix}$

Sincec_(j)(v)∈Λ₁, ∀j∈Z,  (12)we further defineR _(j)(v)={x:Q _(Λ) ₁ (x)=c _(j)(v)}as the Voronoi region associated with c_(j)(v) in Λ₁, and R(v)=U_(j=−∞)^(∞)R_(j)(v). Then

$\begin{matrix}{{{\underset{v \in {\Lambda_{1}/\Lambda_{2}}}{Y}{R_{0}(v)}} = K_{2}},{and}} & (13) \\{{\underset{j = {- \infty}}{\overset{\infty}{Y}}\underset{\;{v \in {\Lambda_{1}/\Lambda_{2}}}}{Y}{R_{j}(v)}} = {{\underset{v \in {\Lambda_{1}/\Lambda_{2}}}{Y}{R(v)}} = {R^{n}.}}} & (14)\end{matrix}$

FIG. 7 illustrates examples of v, C(v) and R(v). The fine lattice pointsare at the centers of the small hexagons. The coarse lattice points areat the centers of the large hexagons. R(v) is the union of the shadedhexagons. The coset C(v) is the set composed of the centers of theshaded hexagons. The fine lattice and coarse lattice may be generated by

${G_{1} = {{\begin{bmatrix}2 & 1 \\0 & \sqrt{3}\end{bmatrix}\mspace{14mu}{and}\mspace{14mu} G_{2}} = \begin{bmatrix}5 & 1 \\\sqrt{3} & {3\sqrt{3}}\end{bmatrix}}},$respectively, and related by

$P = {\begin{bmatrix}2 & {- 1} \\1 & 3\end{bmatrix}.}$

Nested Lattice Quantization.

Throughout this paper, we use the correlation model of X=Y+Z, where X, Yand Z are random vectors in R^(n). X is the source to be coded, Y is theside information, and Z is the noise. Y and Z are independent. In thissection we discuss the performance of nested lattice quantization forgeneral sources where Y and Z are arbitrarily distributed with zeromeans, as well as for the quadratic Gaussian case where Y_(i)˜N(0,σ_(Y)²) and Z_(i)˜N(0,σ_(Z) ²), i=1, 2, . . . , n, are Gaussian. For bothcases, the mean squared error (MSE) is used as the distortionmeasurement.

Zamir et al's nested lattice quantization scheme [18], [16] works asfollows: Let the pseudo-random vector U (also referred to herein as the“dither”), known to both the quantizer encoder and the decoder, beuniformly distributed over the basic Voronoi cell K₁ of the fine latticeΛ₁. For a given target average distortion D, denote α=√{square root over(1−D/σ_(Z) ²)} as the estimation coefficient. Given the realizations ofthe source, the side information and the dither as x, y and u,respectively, then according to [18], the nested quantizer encoderquantizes αx+u to the nearest point x_(Q) _(Λ1) =Q_(Λ) ₁ (αx+u) in Λ₁,computes x_(Q) _(Λ1) −Q_(Λ) ₂ (x_(Q) _(Λ1) ) which is the coset shift ofx_(Q) _(Λ1) with respect to Λ₂, and transmits the index corresponding tothis coset shift.

The nested quantizer decoder receives the index, generates x_(Q) _(Λ1)−Q_(Λ) ₂ (x_(Q) _(Λ1) ) from the index, formsw=x _(Q) _(Λ1) −Q _(Λ) ₂ (x _(Q) _(Λ1) )−u−αyand reconstructs x as {circumflex over (x)}=y+α(w−Q_(Λ) ₂ (w)) usinglinear combination and dithering in estimation.

It is shown in [18] that the Wyner-Ziv R-D function D_(WZ)(R)=σ_(X|Y)²2^(−2R) is achievable with infinite dimensional nested latticequantization for quadratic Gaussian case. In this paper, we analyze thehigh-rate performance of low-dimensional nested lattice quantization,which is of more practical interest as high-dimensional nested latticequantization is too complex to implement, for both general and Gaussiansources.

Our analysis is based on the high-resolution assumption, which means 1)V₁ is small enough so that the PDF of X, ƒ(x), is approximately constantover each Voronoi cell of Λ₁ and 2) dithering can be ignored. With thehigh-rate assumption, α≈1 and the encoder/decoder described abovesimplifies as follows:

(1) The encoder quantizes x to x_(Q) _(Λ1) =Q_(Λ) ₁ (X), computesv=x_(Q) _(Λ1) −Q_(Λ) ₂ (x_(Q) _(Λ1) ), and transmits an indexcorresponding to the coset leader v.

(2) Upon receiving v, the decoder forms w=v−y and reconstructs x as{circumflex over (x)}_(v)=y+W−Q_(Λ) ₂ (w)=v+Q_(Λ) ₂ (y−v).

This simplified nested lattice quantization scheme for high rate isshown in FIG. 8 and was also used in [3].

A. High Rate Performance for General Sources with ArbitraryDistribution.

Theorem 4.1: If a pair of n-D nested lattices (Λ₁,Λ₂) with nesting ratioN=V₂/V₁ is used for nested lattice quantization, the distortion perdimension in Wyner-Ziv coding of X with side information Y at high rateis

$\begin{matrix}{D_{n} = {{{G\left( \Lambda_{1} \right)}V_{1}^{2/n}} + {\frac{1}{n}{{E_{Z}\left\lbrack {{Q_{\Lambda_{2}}(Z)}}^{2} \right\rbrack}.}}}} & (15)\end{matrix}$

The notation ∥a∥ denotes the length (or norm) of the vector a.

Proof: Since

$\begin{matrix}{{R^{n} = {\underset{j = {- \infty}}{\overset{\infty}{Y}}\underset{\;{v \in {\Lambda_{1}/\Lambda_{2}}}}{Y}{R_{j}(v)}}},} & (16)\end{matrix}$the average distortion for a given realization of the side informationY=y is

$\begin{matrix}\begin{matrix}{{D(y)} = {\int_{R^{n}}{{f\left( {x❘y} \right)}{{x - {\hat{x}}_{v}}}^{2}\ {\mathbb{d}x}}}} \\{= {\sum\limits_{v \in {\Lambda_{1}\Lambda_{2}}}{\sum\limits_{j = {- \infty}}^{\infty}{\int_{x \in {R_{j}{(v)}}}{{f\left( {x❘y} \right)}{{x - {\hat{x}}_{v}}}^{2}\ {\mathbb{d}x}}}}}} \\{= {\sum\limits_{v \in {\Lambda_{1}/\Lambda_{2}}}\;{\sum\limits_{j = {- \infty}}^{\infty}\;{\int_{x \in {R_{j}{(v)}}_{\;}}{f\left( {x❘y} \right)}}}}} \\{{{x - {c_{j}(v)} + {c_{j}(v)} - {\hat{x}}_{v}}}^{2}\ {\mathbb{d}x}} \\{= {\sum\limits_{v \in {\Lambda_{1}/\Lambda_{2}}}\;{\sum\limits_{j = {- \infty}}^{\infty}{\int_{x \in {R_{j}{(v)}}}{f\left( {x❘y} \right)}}}}} \\{\begin{bmatrix}\begin{matrix}{{{x - {c_{j}(v)}}}^{2} +} \\{{{{c_{j}(v)} - {\hat{x}}_{v}}}^{2} +}\end{matrix} \\{{2 < {x - {c_{j}(v)}}},{{{c_{j}(v)} - {\hat{x}}_{v}} >}}\end{bmatrix}\ {\mathbb{d}x}} \\{\overset{(a)}{\approx}{\sum\limits_{v \in {\Lambda_{1}/\Lambda_{2}}}\;{\sum\limits_{j = {- \infty}}^{\infty}\begin{bmatrix}{{{f\left( {{c_{j}(v)}❘y} \right)}{\int_{x \in {R_{j}{(v)}}}{{{x - {c_{j}(v)}}}^{2}\ {\mathbb{d}x}}}} +} \\{\int_{x \in {R_{j}{(v)}}}{{f\left( {x❘y} \right)}{{{c_{j}(v)} - {\hat{x}}_{v}}}^{2}\ {\mathbb{d}x}}}\end{bmatrix}}}} \\{\overset{(b)}{=}{\sum\limits_{v \in {\Lambda_{1}/\Lambda_{2}}}\;{\sum\limits_{j = {- \infty}}^{\infty}\begin{bmatrix}\begin{matrix}\begin{matrix}{{{f\left( {{c_{j}(v)}❘y} \right)}{{nG}\left( \Lambda_{1} \right)}V_{1}^{1 + {({2/n})}}} +} \\{\int_{x \in {R_{j}{(v)}}}{{f\left( {x❘y} \right)}{{{Q_{\Lambda_{2}}\left( {c_{j}(v)} \right)} -}}}}\end{matrix} \\{Q_{\Lambda_{2}}\left( {y - {c_{j}(v)} +} \right.}\end{matrix} \\{{\left. {Q_{\Lambda_{2}}\left( {c_{j}(v)} \right)} \right)}^{2}\ {\mathbb{d}x}}\end{bmatrix}}}} \\{\overset{(c)}{\approx}{{{{nG}\left( \Lambda_{1} \right)}V_{1}^{\frac{2}{n}}} + {\sum\limits_{j = {- \infty}}^{\infty}\;{\sum\limits_{v \in {\Lambda_{1}/\Lambda_{2}}}\;\int_{x \in {R_{j}{(v)}}}}}}} \\{{f\left( {x❘y} \right)}{{Q_{\Lambda_{2}}\left( {x - y} \right)}}^{2}\ {\mathbb{d}x}} \\{= {{{nG}\left( \Lambda_{1} \right)V_{1}^{\frac{2}{n}}} + {\int_{x \in R^{n}}{f\left( \ {x❘y} \right)}}}} \\{{{{Q_{\Lambda_{2}}\left( {x - y} \right)}}^{2}\ {\mathbb{d}x}},}\end{matrix} & (17)\end{matrix}$where (a) comes from the high rate assumption and∫_(x∈R) _(j) _((v)) <x−c _(j)(v),c _(j)(v)−{circumflex over (x)} _(v)>dx=0.  (18)

The latter is due to the fact that x−c_(j)(v) is odd spherical symmetricfor x∈R_(j)(v) and both c_(j)(v) and {circumflex over (x)}_(v) are fixedfor x∈R_(j)(v) with given v and y. (b) is due to c_(j)(v)=Q_(Λ) ₁ (x)for x∈R_(j)(v), and{circumflex over (X)} _(v) =c _(j)(v)−Q _(Λ) ₂ (c _(j)(v))+Q _(Λ) ₂ (y−c_(j)(v)+Q _(Λ) ₂ (c _(j)(v)));  (19)and (c) is due toQ _(Λ) ₂ (a+Q _(Λ) ₂ (b))=Q _(Λ) ₂ (a)+Q _(Λ) ₂ (b),∀a,b∈R ^(n)  (20)and the high resolution assumption.

Therefore, the average distortion per dimension over all realizations ofY is

$\begin{matrix}\begin{matrix}{D_{n} = {\frac{1}{n}{E_{Y}\left\lbrack {D(y)} \right\rbrack}}} \\{= {{{G\left( \Lambda_{1} \right)}V_{1}^{2/n}} + {\frac{1}{n}{\int_{x}{\int_{y}{{f\left( {x,y} \right)}{{Q_{\Lambda_{2}}\left( {x - y} \right)}}^{2}\ {\mathbb{d}x}\ {\mathbb{d}y}}}}}}} \\{= {{{G\left( \Lambda_{1} \right)}V_{1}^{2/n}} + {\frac{1}{n}{\int_{y}{{f(y)}{\int_{z}{{f(z)}{{Q_{\Lambda_{2}}(z)}}^{2}\ {\mathbb{d}z}\ {\mathbb{d}y}}}}}}}} \\{= {{{G\left( \Lambda_{1} \right)}V_{1}^{2/n}} + {\frac{1}{n}{{E_{Z}\left\lbrack {{Q_{\Lambda_{2}}(z)}}^{2} \right\rbrack}.}}}}\end{matrix} & (21)\end{matrix}$

Remarks: There are several interesting facts about this rate-distortionfunction. 1) For a fixed pair of the nested lattices (Λ₁,Λ₂), D_(n) onlydepends on Z, i.e., the correlation between X and Y. D_(n) isindependent of the marginal distribution of X (or Y). 2) The first term,G(Λ₁)V₁ ^(2/n), in the expression for D_(n) is due to latticequantization in source coding. It is determined by the geometricstructure and the Voronoi cell volume V₁ of lattice Λ₁. The second term,

${\frac{1}{n}{E_{Z}\left\lbrack {{Q_{\Lambda_{2}}(z)}}^{2} \right\rbrack}},$is the loss due to nesting (or the channel coding component of thenested lattice code). The second term depends on Voronoi cell volume V₂and the distribution of Z. From another point of view, the first term isthe granular component MSE_(g) with respect to the granular lattice Λ₁,and the second term is the overload component MSE_(ol) with respect tothe lattice Λ₂ of the nested quantizer. MSE_(g)=G(Λ₁)V₁ ^(2/n) is thesame as the granular MSE for non-nested lattice quantizer [19].

Corollary 4.1: For the quadratic case, D_(n)→D_(WZ)=σ_(X|Y) ²2^(−2R) asn→∞.

Proof: Since the nested lattice quantizer is a fixed-rate quantizer withthe rate

${R = {\frac{1}{n}{\log\left( \frac{V_{2}}{V_{1}} \right)}}},$then (21) can be rewritten as

$\begin{matrix}{D_{n} = {{{G\left( \Lambda_{1} \right)}V_{2}^{2/n}2^{{- 2}R}} + {\frac{1}{n}{{E_{Z}\left\lbrack {{Q_{\Lambda_{2}}(z)}}^{2} \right\rbrack}.}}}} & (22)\end{matrix}$

For the quadratic Gaussian case, according to equation (3.14) of [18],

$\begin{matrix}{{\frac{1}{n}{\log\left( V_{2} \right)}} \approx {\frac{1}{2}{\log\left( {2\;\pi\; e\;\sigma_{Z}^{2}} \right)}}} & (23)\end{matrix}$

when n is sufficiently large, where σ_(Z) ²=σ_(X|Y) ² is the variance ofthe AWGN Z. Then

$\begin{matrix}\begin{matrix}{{{G\left( \Lambda_{1} \right)}V_{2}^{2/n}2^{{- 2}R}}->{\frac{1}{2\;\pi\; e}2\;\pi\; e\;\sigma_{Z}^{2}2^{{- 2}R}}} \\{= {{\sigma_{Z}^{2}2^{{- 2}\; R}} = {D_{WZ}.}}}\end{matrix} & (24)\end{matrix}$

At the same time, according to equation (3.12) of [18],P_(e)=Pr{Z∉K₂}<∈, with any ∈>0 and sufficiently large n, hence

${\frac{1}{n}{E_{Z}\left\lbrack {{Q_{\Lambda_{2}}(z)}}^{2} \right\rbrack}}->{{0\mspace{14mu}{as}\mspace{14mu} n}->{\infty.}}$Consequently, the performance becomes

$\begin{matrix}\begin{matrix}{D_{n} = {{{G\left( \Lambda_{1} \right)}V_{2}^{2/n}2^{{- 2}R}} + {\frac{1}{n}{E_{Z}\left\lbrack {{Q_{\Lambda_{2}}(z)}}^{2} \right\rbrack}}}} \\{{->{\sigma_{X❘Y}^{2}2^{{- 2}R}}} = D_{WZ}}\end{matrix} & (25)\end{matrix}$as n→∞, for the quadratic Gaussian case. This limit agrees with thestatement in [18], which claims that the nested lattice quantization canachieve the Wyner-Ziv limit asymptotically as the dimensionality goes toinfinity.

B. A Lower Bound of the Performance for Quadratic Case.

The source-coding-loss in (21) has an explicit form, while thechannel-coding loss is not so directly expressed. Among all the possiblepatterns of the additive channels, AWGN is of most interest. In suchcase Z is a Gaussian variable with zero mean and variance σ_(Z)²=σ_(X|Y) ². From Theorem 4.1, we obtain a lower bound of the high-rateR-D performance of low-dimensional nested lattice quantizers forWyner-Ziv coding, when Z is Gaussian, stated as the following corollary.

Corollary 4.2: For X=Y+Z, Y˜N(0,σ_(Y) ²) and Z˜N(0,σ_(Z) ²), the R-Dperformance of Wyner-Ziv coding for X with side information Y using n-Dnested lattice quantizers is lower-bounded at high rate by

$\begin{matrix}{{{D_{n}(R)} \geq {{\overset{\_}{D}}_{n}(R)} \equiv {\min\limits_{V_{2} > 0}{\delta_{n}(R)}}},} & (26) \\{where} & \; \\{{{\delta_{n}(R)} \equiv {{G_{n}V_{2}^{2/n}2^{{- 2}R}} + {\frac{1}{n}{\sum\limits_{j = 1}^{\infty}\;{\left( {\left( {{2\; j} - 1} \right)\gamma_{n}V_{2}^{2/n}} \right){u\left( \frac{j^{2}V_{2}^{2/n}{\Gamma\left( {\frac{n}{2} + 1} \right)}^{2/n}}{2\;\pi\;\sigma_{Z}^{2}} \right)}}}}}},} & (27)\end{matrix}$γ_(n) is the n-D Hermite's constant [13], [26], and u(t) is defined in[26] as

$\begin{matrix}{{u(t)} = {\begin{Bmatrix}{{\mathbb{e}}^{- t}\left( {1 + \frac{t}{1!} + \frac{t^{2}}{2!} + K + \frac{t^{\frac{n}{2} - 1}}{\left( {\frac{n}{2} - 1} \right)!}} \right)} & {{if}\mspace{14mu} n\mspace{14mu}{is}\mspace{14mu}{even}} \\{{\mathbb{e}}^{- t}\left( {1 + \frac{t^{1/2}}{\left( {1/2} \right)!} + \frac{t^{3/2}}{\left( {3/2} \right)!} + \Lambda + \frac{T^{\frac{n}{2} - 1}}{\left( {\frac{n}{2} - 1} \right)!}} \right)} & {{if}\mspace{14mu} n\mspace{14mu}{is}\mspace{14mu}{odd}}\end{Bmatrix}.}} & (28)\end{matrix}$

Specifically, when n=1, the best possible high rate performance is

$\begin{matrix}{{{D_{1}(R)} = {\min\limits_{V_{2} > 0}\left\{ {{G_{1}V_{2}^{2}2^{{- 2}R}} + {V_{2}^{2}{\sum\limits_{j = 0}^{\infty}{\left( {{2\; j} + 1} \right){Q\left( {\frac{V_{2}}{\sigma_{Z}}\left( {j + \frac{1}{2}} \right)} \right)}}}}} \right\}}},} & (29) \\{where} & \; \\{{Q(t)} = {\frac{1}{\sqrt{2\;\pi}}{\int_{t}^{\infty}{{\mathbb{e}}^{{- \tau^{2}}/2}\ {{\mathbb{d}\tau}.}}}}} & (30)\end{matrix}$

Proof: 1) Rate Computation: Note that the nested lattice quantizer is afixed rate quantizer with rate

$R = {\frac{1}{n}{{\log_{2}\left( \frac{V_{2}}{V_{1}} \right)}.}}$

2) Distortion computation: Define

$\begin{matrix}{{L_{2} = {\frac{1}{2}{\min\limits_{{\forall l},{l^{\prime} \in \Lambda_{2}},{l \neq l^{\prime}}}{{l - l^{\prime}}}}}},} & (31) \\{and} & \; \\{{P_{Z}(L)} = {{\Pr\left( {{Z} > L} \right)}.}} & (32)\end{matrix}$

For the 1-D (scalar) case, P_(Z) can be expressed in terms of the Qfunction and E_(Z)[∥Q_(Λ) ₂ (z)∥²] simplifies to [27]

$\begin{matrix}{{E_{Z}\left\lbrack {{Q_{\Lambda_{2}}(z)}}^{2} \right\rbrack} = {V_{2}^{2}{\sum\limits_{j = 0}^{\infty}{\left( {{2\; j} + 1} \right){{Q\left( {\frac{V_{2}}{\sigma_{z}}\left( {j + \frac{1}{2}} \right)} \right)}.}}}}} & (33)\end{matrix}$

For the n-D (with n>1) case, note that [28]L ₂ ²=γ(Λ₂)V(Λ₂)^(2/n),  (34)and∥Q _(Λ) ₂ (z)∥² ≧∥z∥ ² −∥z−Q _(Λ) ₂ (z)∥² ≧∥z∥ ² −L ₂ ²,  (35)where γ(Λ₂) is the Hermite's constant of lattice Λ₂ [13], [26].

Then we get

$\begin{matrix}\begin{matrix}{{E_{Z}\left\lbrack {{Q_{A_{2}}(z)}}^{2} \right\rbrack} = {{\sum\limits_{j = 1}^{x}\;{\int_{({{{({j - 1})}L_{2}} < {z} \leq {jL}_{2}})}{{f(z)}{{Q_{\Lambda_{2}}(z)}}^{2}\ {\mathbb{d}z}}}} \geq}} \\{{\sum\limits_{j = 1}^{\infty}{\int_{({{{({j - 1})}L_{2}} < {z} \leq {jL}_{2}})}{{f(z)}\left( {{z}^{2} - L_{2}^{2}} \right)\ {\mathbb{d}z}}}} \geq} \\{\sum\limits_{j = 1}^{\infty}{\int_{({{{({j - 1})}L_{2}} < {z} \leq {jL}_{2}})}{{f(z)}\left( {{\left( {j - 1} \right)^{2}L_{2}^{2}} - L_{2}^{2}} \right)\ {\mathbb{d}z}}}} \\{= {\sum\limits_{j = 1}^{\infty}{\left( {{\left( {j - 1} \right)^{2}L_{2}^{2}} - L_{2}^{2}} \right)\left\lbrack {{P_{Z}\left( {\left( {j - 1} \right)L_{2}} \right)} - {P_{Z}\left( {jL}_{2} \right)}} \right\rbrack}}} \\{= {\sum\limits_{j = 1}^{\infty}{\left( {\left( {{2\; j} - 1} \right)L_{2}^{2}} \right){P_{Z}\left( {jL}_{2} \right)}}}} \\{{= {\sum\limits_{j = 1}^{\infty}{\left\lbrack {\left( {{2\; j} - 1} \right){\gamma\left( \Lambda_{2} \right)}V_{2}^{\frac{2}{n}}} \right\rbrack{P_{e}\left( \frac{j^{2}V_{2}^{\frac{2}{n}}{\Gamma\left( {\frac{n}{2} + 1} \right)}^{\frac{2}{n}}}{2\;\pi\;\sigma_{Z}^{2}} \right)}}}},}\end{matrix} & (36)\end{matrix}$where

Γ(t) = ∫₀^(∞)u^(t − 1)𝕖^(−u) 𝕕uis Euler's gamma function, and P_(e)(·) is defined in [26] as the symbolerror probability under maximum likelihood decoding while transmittingthe lattice points over an AWGN channel. A lower bound of P_(e)(·) wasalso given in [26] as P_(e)(t)≧u(t).

Then Theorem 4.1 and (36) give

$\begin{matrix}{D_{n} \geq \delta_{n} \equiv {{G_{n}V_{1}^{2/n}} + {\frac{1}{n}{\sum\limits_{j = 1}^{\infty}{\left( {\left( {{2\; j} - 1} \right)\gamma_{n}V_{2}^{2/n}} \right){{u\left( \frac{j^{2}V_{2}^{2/n}{\Gamma\left( {\frac{n}{2} + 1} \right)}^{2/n}}{2\;\pi\;\sigma_{Z}^{2}} \right)}.}}}}}} & (37)\end{matrix}$

Using

${R = {\frac{1}{n}{\log_{2}\left( \frac{V_{2}}{V_{1}} \right)}}},$we eliminate V₁ in D_(n) and obtain a lower bound on D_(n)(R) as

$\begin{matrix}{{{D_{n}(R)} \geq {{\overset{\_}{D}}_{n}(R)} \equiv {\min\limits_{V_{2} > 0}{\delta_{n}(R)}}},} & (38) \\{where} & \; \\{{\delta_{n}(R)} = {{G_{n}V_{2}^{2/n}2^{{- 2}R}} + {\frac{1}{n}{\sum\limits_{j = 1}^{\infty}{\left( {\left( {{2\; j} - 1} \right)\gamma_{n}V_{2}^{2/n}} \right){{u\left( \frac{j^{2}V_{2}^{2/n}{\Gamma\left( {\frac{n}{2} + 1} \right)}^{2/n}}{2\;\pi\;\sigma_{Z}^{2}} \right)}.}}}}}} & (39)\end{matrix}$

FIG. 9 shows δ₂(R) with different V₂'s using nested A₂ lattices (i.e.,hexagonal lattices) in 2-D with σ_(Z) ²=0.01. The lower bound D ₂(R) isthe lower convex hull of all operational R-D points with different V₂,as shown in FIG. 10. We observe from FIG. 10 that the gap from D _(n)(R)to D_(WZ)(R) in dBs keeps increasing as the rate increases with σ_(Z)²=0.01. This increasing gap comes from the fact that, the granular MSEcomponent

${{MSE}_{g} \equiv {G_{n}V_{1}^{2/n}}} = {{\frac{1}{12\gamma_{g}}\left( \frac{V_{2}}{N} \right)^{2/n}} = {\frac{1}{12\gamma_{g}}V_{2}^{2/n}2^{{- 2}R}}}$is away from the benchmark 2^(−2R) with an increasing gap as V₂increases, where

$\gamma_{g} \equiv \frac{1/12}{G_{n}}$is the granular gain [19] of lattice Λ₁, and

${R = {\frac{1}{n}\log\; N}},$N is the nesting ratio, as shown in FIG. 11.

FIG. 12 plots D _(n)(R) for n=1, 2, 4, 8 and 24 with σ_(Z) ²=0.01. Wesee that D _(n)(R) gets closer and closer to the Wyner-Ziv R-D functionD_(WZ)(R)=σ_(X|Y) ²2^(−2R) as n goes to infinity.

C. Discussion of the Correlation-Asymptotical Property.

As to the asymptotical property of the nested-lattice quantization forWyner-Ziv coding, we have the following statement. Here asymptoticalmeans that the correlation

$\rho \equiv \frac{E\lbrack{XY}\rbrack}{\sqrt{{E\left\lbrack X^{2} \right\rbrack}{E\left\lbrack Y^{2} \right\rbrack}}}$between the source X and the side information Y goes to 1asymptotically. If we fix σ_(Y) ², then the asymptotical performance isthe one when σ_(Z) ²→0.

Corollary 4.3: The distortion of the nested lattice quantizationmaintains a constant gap (in dB) to the Wyner-Ziv bound for all 0<σ_(Z)²<1.

Proof: Denote

${s = V_{2}^{2/n}},{t = {\frac{j^{2}{\Gamma\left( {\frac{n}{2} + 1} \right)}^{2/n}}{2{\pi\sigma}_{Z}^{2}}s}},$From Corollary 4.1, we get

$\begin{matrix}{\delta = {{G_{n}s\; 2^{{- 2}R}} + {\frac{1}{n}{\sum\limits_{j = 1}^{\infty}{\left( {\left( {{2j} - 1} \right)\gamma_{n}s} \right){{u(t)}.}}}}}} & (40)\end{matrix}$

Fix rate R and dimensionality n (without loss of generality, assume n iseven), and minimize δ with respect to s,

$\begin{matrix}{\frac{\mathbb{d}\delta}{\mathbb{d}s} = {{G_{n}2^{{- 2}R}} + {\frac{1}{n}{\sum\limits_{j = 1}^{\infty}\left( {\left( {{2j} - 1} \right)\gamma_{n}{u(t)}} \right)}} + {\frac{1}{n}{\sum\limits_{j = 1}^{\infty}\left( {\left( {{2j} - 1} \right)\gamma_{n}t\frac{\mathbb{d}{u(t)}}{\mathbb{d}t}} \right)}}}} & (41) \\{where} & \; \\\begin{matrix}{\frac{\mathbb{d}{u(t)}}{\mathbb{d}t} = {{- {{\mathbb{e}}^{- t}\left( {1 + \frac{t}{1!} + K + \frac{t^{\frac{n}{2} - 1}}{\left( {\frac{n}{2} - 1} \right)!}} \right)}} +}} \\{{\mathbb{e}}^{- t}\left( {1 + \frac{t}{1!} + K + \frac{t^{\frac{n}{2} - 2}}{\left( {\frac{n}{2} - 2} \right)!}} \right)} \\{= {{- {\mathbb{e}}^{- t}}\frac{t^{\frac{n}{2} - 1}}{\left( {\frac{n}{2} - 1} \right)!}}}\end{matrix} & (42) \\{then} & \; \\{{\frac{\mathbb{d}\delta}{\mathbb{d}s}G_{n}2^{{- 2}R}} + {\frac{1}{n}{\sum\limits_{j = 1}^{\infty}\left( {\left( {{2j} - 1} \right)\gamma_{n}{u(t)}} \right)}} - {\frac{1}{n}{\sum\limits_{j = 1}^{\infty}{\left( {\left( {{2j} - 1} \right)\gamma_{n}t\;{\mathbb{e}}^{- t}\frac{t^{{({n/2})} - 1}}{\left( {\frac{n}{2} - 1} \right)!}} \right).}}}} & (43)\end{matrix}$

Set

${\frac{\mathbb{d}\delta}{\mathbb{d}s} = 0},$and denote the optimal s as s₀, and denote the corresponding t as

${t_{0} = {\frac{j^{2}A}{2\;\pi\;\sigma_{z}^{2}}s_{0}}},$we get

$\begin{matrix}{{{{G_{n}2^{{- 2}R}} + {\frac{1}{n}{\sum\limits_{j = 1}^{\infty}\left( {\left( {{2\; j} - 1} \right)\gamma_{n}{u\left( t_{0} \right)}} \right)}}} = {\frac{1}{\; n}{\sum\limits_{j\; = \; 1}^{\;\infty}\left( {\left( {{2\; j} - 1} \right)\gamma_{\; n}t_{\; 0}{\mathbb{e}}^{- \; t_{\; 0}}\frac{\; t_{\; 0}^{{({n/2})}\; - \; 1}}{\left( \;{\frac{n}{\; 2}\; - \; 1} \right)!}} \right)}}},{hence}} & (44) \\\begin{matrix}{\delta^{*} = {{G_{n}s_{0}2^{{- 2}R}} + {\frac{1}{n}{\sum\limits_{j = 1}^{\infty}\left( {\left( {{2\; j} - 1} \right)\gamma_{n}s_{0}{u\left( t_{0} \right)}} \right)}}}} \\{= {\frac{1}{2}{\sum\limits_{j = 1}^{\infty}\left( {\left( {{2\; j} - 1} \right)\gamma_{n}s_{0}t_{0}{\mathbb{e}}^{- t_{0}}\frac{t_{0}^{{({n/2})} - 1}}{\left( {\frac{n}{2} - 1} \right)!}} \right)}}} \\{= {\frac{1}{n}{\sum\limits_{j = 1}^{\infty}{\left( {\left( {{2\; j} - 1} \right)\gamma_{n}\frac{2\;\pi\;\sigma_{z}^{2}}{j^{2}A}t_{0}^{2}{\mathbb{e}}^{- t_{0}}\frac{t_{0}^{{({n/2})} - 1}}{\left( {\frac{n}{2} - 1} \right)!}} \right).}}}}\end{matrix} & (45)\end{matrix}$

From (45) one can see that the optimal to only depends on the rate R andthe dimensionality n. The optimal t₀ stays unchanged with differentσ_(Z) ², thus the optimized distortion δ* is a linear function of σ_(Z)², denoted as D=δ*=B(R,n)σ_(Z) ². Since the Wyner-Ziv bound isD_(WZ)=σ_(Z) ²2^(−2R), the gap (in terms of dB) from the practicaloptimized distortion D to Wyner-Ziv bound D_(WZ) with fixed R and n is

$\begin{matrix}{{\Delta\; D} = {{10\log_{10}\frac{D}{D_{WZ}}} = {10\log_{10}\frac{B\left( {R,n} \right)}{2^{{- 2}R}}}}} & (46)\end{matrix}$which stays constant for all σ_(Z) ²<1.

This result verifies our simulation results which show that thedistortion of the nested lattice quantizer does NOT approach theWyner-Ziv bound as the correlation between the source and the sideinformation goes to 1 asymptotically.

Slepian-Wolf Coded Nested Lattice Quantization (SWC-NQ).

In this section, we evaluate the boundary gain of the source codingcomponent of nested lattice quantization. Motivated by this evaluation,we introduce SWC-NQ and analyze its performance.

A. Motivation of SWC-NQ.

From Theorem 4.1, the distortion per dimension of the nested latticequantizer is D_(n)=MSE_(g)+MSE_(ol), where MSE_(g)=G(Λ₁)V₁ ^(2/n) is thegranular component of the distortion, characterized by the granular gain

${{\gamma_{g}\left( \Lambda_{1} \right)} = \frac{1/12}{G\left( \Lambda_{1} \right)}},$while

${MSE}_{ol} = {\frac{1}{n}{E_{z}\left\lbrack {{Q_{\Lambda_{2}}(z)}}^{2} \right\rbrack}}$is the overload component of the distortion, characterized by theboundary gain γ_(b)(Λ₂). The boundary gain γ_(b)(Λ₂) is defined in [19]as follows. Suppose that Λ₂ is the boundary (coarse) lattice with itsVoronoi region as the n-dimensional support region, and it has the sameoverload probability as a cubic support region of size αM centered atthe origin. The boundary gain is then defined as the ratio of thenormalized volume (αM)² of the cubic support region to the normalizedvolume V₂ ^(2/n), as

$\begin{matrix}{{\gamma_{b}\left( \Lambda_{2} \right)} = {\frac{\left( {\alpha\; M} \right)^{2}}{V_{2}^{2/n}}.}} & (47) \\{Since} & \; \\\begin{matrix}{{MSE}_{g} = {{G\left( \Lambda_{1} \right)}V_{1}^{2/n}}} \\{= {\frac{1}{12{\gamma_{g}\left( \Lambda_{1} \right)}}V_{2}^{2/n}N^{{- 2}/n}}} \\{= {\frac{1}{12{\gamma_{g}\left( \Lambda_{1} \right)}}\frac{\alpha\; M}{\gamma_{b}\left( \Lambda_{2} \right)}N^{{- 2}/n}}}\end{matrix} & (48)\end{matrix}$

If the nesting ratio N stays constant (i.e., the codebook size N staysconstant), then MSE_(g) will be reduced by a factor of γ_(b)(Λ₂),without affecting MSE_(ol) because the overload probability staysunchanged.

To increase the boundary gain γ_(b)(Λ₂), a second-stage of binning canbe applied to the quantization indices. The essence of binning is achannel code which partitions the support region into several cosets.Assuming the channel code is strong enough so that there is no extraoverload probability introduced (i.e., it is lossless coding for theindices without decoding error), and the channel code partitions thesupport region K₂ into m cosets, with the set composed of the cosetleaders denoted as S, then #(S)=m and S is the support region for thequantization indices and hence the support region for the nestedquantization, with Vol(S)=(m/N)V₂<V₂. Then the effective volume of thesupport region decreases by a factor of the coset size after the secondstage of binning, and therefore, the boundary gain γ_(b)(Λ₂) increases.The notation “#(A)” denotes the cardinality of (i.e., the number ofelements in) the set A.

We thus propose a framework for Wyner-Ziv coding of i.i.d. sources basedon SWC-NQ, which involves nested quantization (NQ) and Slepian-Wolfcoding (SWC). The SWC operates as the second binning scheme. Despite thefact that there is almost no correlation among the nested quantizationindices that identify the coset leaders v∈Λ₁/Λ₂ of the pair of nestedlattices (Λ₁, Λ₂), there still remains correlation between v and theside information Y. Ideal SWC can be used to compress v to the rate ofR=H(v|Y). State-of-the-art channel codes, such as LDPC codes, can beused to approach the Slepian-Wolf limit H(v|Y) [29]. The role of SWC inSWC-NQ is to exploit the correlation between v and Y for furthercompression.

B. Uniform High Rate Performance.

Let's evaluate the high rate performance for the quadratic Gaussian casefirst. For this case, a lower bound for the high-rate performance ofSWC-NQ with a pair of arbitrary nested lattices (Λ₁,Λ₂) is given as

$\begin{matrix}{{D_{n}(R)} \geq {{{G\left( \Lambda_{1} \right)}2^{{({2/n})}{h^{\prime}{({X,\Lambda_{2}})}}}\sigma_{X❘Y}^{2}2^{{- 2}R}} + {\frac{1}{n}{\sum\limits_{j = 1}^{\infty}{\left( {\left( {{2j} - 1} \right)\gamma_{n}V_{2}^{2/n}} \right){u\left( \frac{j^{2}V_{2}^{2/n}{\Gamma\left( {\frac{n}{2} + 1} \right)}^{2/n}}{2{\pi\sigma}_{Z}^{2}} \right)}}}}}} & (49) \\{where} & \; \\{{{h^{\prime}\left( {X,\Lambda_{2}} \right)} \equiv {- {\int_{x \in R^{n}}{{\overset{\_}{f}(x)}{\log_{2}\left\lbrack {\sum\limits_{i = {- \infty}}^{\infty}{\overset{\_}{f}\left( {x + \frac{c_{i}(0)}{\sigma_{X|Y}}} \right)}} \right\rbrack}\ {\mathbb{d}x}}}}},} & (50)\end{matrix}$ƒ(·) is the PDF of an n-D i.i.d. Gaussian source with 0 mean and unitvariance, u(t) is defined in (28), and c_(i)(0) is defined above (in thesection entitled “Lattices and Nested Lattices”), as the lattice pointsof Λ₂.

Proof: The proof to this lower bound is provided later.

For example, the lower bounds of D(R) for the 1-D case with different V₂are plotted in FIG. 13.

FIG. 13 gives us a hint that, intuitively, the best R-D function ofSWC-NQ is the R-D function as if the side information were alsoavailable at the encoder, and maintains a constant gap of 2πeG_(n) fromthe Wyner-Ziv limit in dB. Here the best means that, for a givendistortion D, the minimal achievable rate R over all possible V₂, orequivalently, the minimal achievable distortion D over all possible V₂for a given rate R. This claim is stated and proved as follows. Let'sstart with the following lemma and then prove the main theorem.

Lemma 5.1: For nested lattice quantization, denote W≡Q_(Λ) ₁ (X), andV≡W−Q_(Λ) ₂ (W). At high rate, H(V|Y)≈H(W|Y).

Proof: The proof is provided later.

Theorem 5.2: The optimal R-D performance of SWC-NQ for general sourcesusing low-dimensional nested lattices for Wyner-Ziv coding at high rateis

$\begin{matrix}{{{D_{n}^{*}(R)} \equiv {\min\limits_{V_{2}}{D(R)}}} = {G_{n}2^{{({2/n})}{h{({X|Y})}}}{2^{{- 2}R}.}}} & (51)\end{matrix}$

Proof: 1) When V₂→∞, Q_(Λ) ₂ (Q_(Λ) ₁ (X))=0 and Q_(Λ) ₂ (Z)=0, then

$\begin{matrix}\begin{matrix}{{nR} = {H\left( V \middle| Y \right)}} \\{= {H\left( {{Q_{\Lambda_{1}}(X)} - {Q_{\Lambda_{2}}\left( {Q_{\Lambda_{1}}(X)} \right)}} \middle| Y \right)}} \\{= {H\left( {Q_{\Lambda_{1}}(X)} \middle| Y \right)}} \\{= {{h\left( X \middle| Y \right)} - {\log\left( V_{1} \right)}}}\end{matrix} & (52)\end{matrix}$and D_(n)(R)=G_(n)V₁ ^(2/n). Combine R and D_(n) through V₁ and we getthe R-D function as

$\begin{matrix}{{D_{n}(R)}_{V_{2}\rightarrow\infty} = {G_{n}2^{{({2/n})}{h{({X|Y})}}}{2^{{- 2}R}.}}} & (53) \\{{{{Since}\mspace{14mu}{D_{n}^{*}(R)}} \equiv {\min\limits_{V_{2}}{D(R)}} \leq {D_{n}(R)}_{V_{2}\rightarrow\infty}},{{{then}\mspace{14mu}{D_{n}^{*}(R)}} \leq {G_{n}2^{{({2/n})}{h{({X|Y})}}}{2^{{- 2}R}.}}}} & (54)\end{matrix}$

2) Denote w≡Q_(Λ) ₁ (x), and S₁≡{(x,{circumflex over(x)}):E[d(x,{circumflex over (x)})]≦D}. The rate of Wyner-Ziv codingwith respect to a given distortion D is [1]

$\begin{matrix}\begin{matrix}{{{nR}^{*}(D)} = {\min\limits_{{p{(v)}},{p{({{\hat{x}|v},y})}},{{({x,\hat{x}})} \in S_{1}}}{I\left( {X;\left. V \middle| Y \right.} \right)}}} \\{\overset{(a)}{=}{\min\limits_{{p{(v)}},{p{({{\hat{x}|v},y})}},{{({x,\hat{x}})} \in S_{1}}}{H\left( V \middle| Y \right)}}} \\{{\overset{(b)}{\approx}{\min\limits_{{p{(v)}},{p{({{\hat{x}|v},y})}},{{({x,\hat{x}})} \in S_{1}}}{H\left( W \middle| Y \right)}}},}\end{matrix} & (55)\end{matrix}$where (a) comes from H(V|X,Y)=0 and (b) comes from Lemma 5.1.

Define S₂≡{(x,{circumflex over (x)}) E[d(x,w)]≦D}. From Theorem 4.1,

$\begin{matrix}\begin{matrix}{{E\left\lbrack {d\left( {x,\hat{x}} \right)} \right\rbrack} = {{G_{n}V_{1}^{2/n}} + {\frac{1}{n}{E_{Z}\left\lbrack {{Q_{\Lambda_{2}}(z)}}^{2} \right\rbrack}}}} \\{= {{{E\left\lbrack {d\left( {x,w} \right)} \right\rbrack} + {\frac{1}{n}{E_{Z}\left\lbrack {{Q_{\Lambda_{2}}(z)}}^{2} \right\rbrack}}} \geq}} \\{{E\left\lbrack {d\left( {x,w} \right)} \right\rbrack}.}\end{matrix} & (56)\end{matrix}$

Then ∀(x,{circumflex over (x)})∈S₁,D≧E[d(x,{circumflex over (x)})]≧E[d(x,w)],  (57)it means that (x,{circumflex over (x)})∈S₂.

Then S₁ ⊂S₂, and

$\begin{matrix}{{{nR}^{*}(D)} \approx {\min\limits_{{p{(v)}},{p{({{\hat{x}|v},y})}},{{({x,\hat{x}})} \in S_{1}}}{H\left( W \middle| Y \right)}} \geq {\min\limits_{{p{(v)}},{p{({{\hat{x}|v},y})}},{{({x,\hat{x}})} \in S_{2}}}{{H\left( W \middle| Y \right)}.}}} & (58)\end{matrix}$

Since H(W|Y)=h(X|Y)−log(V₁) and E[d(x,{circumflex over (x)})|Y]=G_(n)V₁^(2/n), R*(D) can be calculated using Lagrangian method, as

$\begin{matrix}{{{{nR}^{*}(D)} \geq {\min\limits_{{p{(v)}},{p{({{\hat{x}|v},y})}},{{({x,\hat{x}})} \in S_{2}}}{H\left( W \middle| Y \right)}}} = {{h\left( X \middle| Y \right)} - {\frac{n}{2}{{\log\left( \frac{D}{G_{n}} \right)}.}}}} & (59) \\{Then} & \; \\{{D_{n}^{*}(R)} \geq {G_{n}2^{{({2/n})}{h{({X|Y})}}}{2^{{- 2}R}.}}} & (60)\end{matrix}$

From (54) and (60), it is proved that, at high rate, the best R-Dfunction of SWC-NQ using low-dimensional lattices isD _(n)*(R)=G _(n)2^((2/n)h(X|Y))2^(−2R).  (61)

Corollary 5.4: The optimal R-D performance of quadratic Gaussian SWC-NQusing low-dimensional nested lattices at high rate isD _(n)*(R)=G _(n)2^((2/n)h(X|Y))2^(−2R).  (62)

We thus conclude that at high rates, SWC-NQ performs the same as thetraditional entropy-constrained lattice quantization with the sideinformation available at both the encoder and decoder. Specifically, theR-D functions with 1-D (scalar) lattice and 2-D (hexagonal) lattice are1.53 dB and 1.36 dB away from the Wyner-Ziv bound, respectively.

Remarks: We found that for finite rate R and small n (e.g., n=1 and 2),the optimal V₂, denoted as V₂*, that minimizes the distortion D_(n)(R)is also finite. FIGS. 14( a) and (b) plot the optimal V₂* (scaled byσ_(Z)) as a function of R for the 1-D (n=1) and 2-D (n=2) case. We seethat as R goes to infinity, V₂* also goes to infinity. We also observethat for fixed R and n, D_(n)(R) stays roughly constant for V₂>V₂*.

Code Design and Simulation Results.

In this section, the optimal decoder for nested quantizer at low rate isintroduced, and the issue of code design is also discussed, along withsimulation results.

A. The Optimal Decoder for Nested Quantizer at Low Rate.

The optimal estimator for the decoder corresponding to the nestedquantizer should minimize the distortion between X and the reconstructed{circumflex over (X)}. If mean squared error (MSE) is used as thedistortion measure, {circumflex over (x)} will be E[X|j, y], where j isthe received bin index corresponding to the coset leader v=x_(Q) _(Λ1)−Q_(Λ) ₂ (x_(Q) _(Λ1) ). Let Y and Z be independent zero mean Gaussianrandom variables with variances σ_(Y) ² and σ_(Z) ², then we haveX|y˜N(y,σ_(z) ²).

When n=1, the optimal decoder for nested quantizer can be deriveddirectly from E[X|j,y] as

$\begin{matrix}{{{\hat{x}\left( {j,y} \right)} = {\frac{1}{\sqrt{2{\pi\sigma}_{Z}^{2}}}{\sum\limits_{n = {- \infty}}^{\infty}{\int_{{jq} + {nQ}}^{{{({j + 1})}q} + {nQ}}{x\;{\exp\left( {- \frac{{{x - y}}^{2}}{2\sigma_{Z}^{2}}} \right)}\ {\mathbb{d}x}}}}}},} & (63)\end{matrix}$where q and Q are the uniform intervals of the two nested lattices usedby the scalar quantizer, with

${\frac{Q}{q} = N},$where N is the nesting ratio. At high rates, the rate distortionperformance using this non-linear estimation matches our analysis in(29); at low rate, such estimation method helps to boost theperformance.

When n>1, the optimal decoder for the nested quantizer is stated asfollows.

Theorem 6.3: The optimal decoder for the nested quantizer in the senseof MSE is

$\begin{matrix}{\hat{x} = {{E\left\lbrack {\left. x \middle| y \right.,j} \right\rbrack} = {\frac{\int_{R{(v)}}^{\;}{x\;{f\left( x \middle| y \right)}\ {\mathbb{d}x}}}{\int_{R{(v)}}^{\;}{{f\left( x \middle| y \right)}\ {\mathbb{d}x}}}.}}} & (64)\end{matrix}$

Proof

$\begin{matrix}\begin{matrix}{\hat{x} = {E\left\lbrack {{x❘y},j} \right\rbrack}} \\{= {\int_{R^{n}}{{{xf}\left( {{x❘y},j} \right)}\ {\mathbb{d}x}}}} \\{= {\int_{R^{n}}{x\frac{f\left( {x,{j❘y}} \right)}{P\left( {j❘y} \right)}\ {\mathbb{d}x}}}} \\{= \frac{\int_{R^{n}}{{{xf}\left( {x,{j❘y}} \right)}\ {\mathbb{d}x}}}{P\left( {j❘y} \right)}} \\{= {\frac{\int_{R^{n}}{{{xf}\left( {x❘y} \right)}{P\left( {{j❘x},y} \right)}\ {\mathbb{d}x}}}{\int_{R{(v)}}{{f\left( {x❘y} \right)}\ {\mathbb{d}x}}}.}}\end{matrix} & (65)\end{matrix}$

Note that j, x, y form a Markov chain as y

x

j , then

$\begin{matrix}{{{P\left( {\left. j \middle| x \right.,y} \right)} = {{P\left( j \middle| x \right)}\begin{Bmatrix}{{0\mspace{14mu}{if}\mspace{14mu} x} \notin {R(v)}} \\{{1\mspace{14mu}{if}\mspace{14mu} x}\; \in \;{R(v)}}\end{Bmatrix}}},} & (66)\end{matrix}$and we get

$\begin{matrix}\begin{matrix}{\hat{x} = \frac{\int_{R^{n}}^{\;}{x\;{f\left( x \middle| y \right)}{P\left( j \middle| x \right)}\ {\mathbb{d}x}}}{\int_{R{(v)}}^{\;}{{f\left( x \middle| y \right)}\ {\mathbb{d}x}}}} \\{= {\frac{\int_{R{(v)}}^{\;}{x\;{f\left( x \middle| y \right)}\ {\mathbb{d}x}}}{\int_{R{(v)}}^{\;}{{f\left( x \middle| y \right)}\ {\mathbb{d}x}}}.}}\end{matrix} & (67)\end{matrix}$

Estimation at the decoder plays an important role for low-rateimplementation. We thus apply an optimal non-linear estimator at thedecoder at low rates in our simulations.

Corollary 6.5: The optimal estimator stated in Theorem 6.3 degeneratesto the linear one {circumflex over (x)}=v+Q_(Λ) ₂ (y−v) at high rates aswe discussed above in the section entitled “Nested Lattice Quantization”and in the section entitled “Slepian-Wolf Coded Nested LatticeQuantization”.

Proof: At high rate,

$\begin{matrix}\begin{matrix}{\hat{x} = \frac{\int_{R{(v)}}^{\;}{x\;{f\left( x \middle| y \right)}\ {\mathbb{d}x}}}{\int_{R{(v)}}^{\;}{{f\left( x \middle| y \right)}\ {\mathbb{d}x}}}} \\{= \frac{\sum\limits_{i = {- \infty}}^{\infty}\;{\int_{R_{i}{(v)}}^{\;}{x\;{f\left( x \middle| y \right)}\ {\mathbb{d}x}}}}{\sum\limits_{i = {- \infty}}^{\infty}\;{\int_{R_{i}{(v)}}^{\;}{{f\left( x \middle| y \right)}\ {\mathbb{d}x}}}}} \\{\overset{(a)}{\approx}\frac{\sum\limits_{i = {- \infty}}^{\infty}\;{{c_{i}(v)}{\int_{R_{i}{(v)}}^{\;}\;{{f\left( x \middle| y \right)}\ {\mathbb{d}x}}}}}{\sum\limits_{i = {- \infty}}^{\infty}\;{\int_{R_{i}{(v)}}^{\;}{{f\left( x \middle| y \right)}\ {\mathbb{d}x}}}}} \\{\overset{(b)}{\approx}\frac{v + {{Q_{\Lambda_{2}}\left( {y - v} \right)}{\int_{K{({v + {Q_{\Lambda_{2}}{({y - v})}}})}}^{\;}{{f\left( x \middle| y \right)}\ {\mathbb{d}x}}}}}{\int_{K{({v + {Q_{\Lambda_{2}}{({y - v})}}})}}^{\;}{{f\left( x \middle| y \right)}\ {\mathbb{d}x}}}} \\{{= {v + {Q_{\Lambda_{2}}\left( {y - v} \right)}}},}\end{matrix} & (68)\end{matrix}$which is the linear estimator, discussed above in the section entitled“Nested Lattice Quantization” and in the section entitled “Slepian-WolfCoded Nested Lattice Quantization”, for high rate performance. Steps (a)and (b) of (68) come from the high rate assumption.

Since the non-linear estimation is a definite integral of a simplefunction over a disconnected region which includes many isolated Voronoicells, we choose the Monte Carlo method to do this integral. In onesimulation, for each scaling factor, there are totally 10⁴×10⁴=10⁸ pairsof {x,y} to be simulated, and for each pair of {x,y}, there are 10⁴samples to calculate this definite integral.

FIG. 15 shows the improvement gained by using the optimal (non-linear)estimator at low rates, for n=2 and σ_(Z) ²=0.01.

B. Code Design of LDPC Codes.

Let J (0≦J≦N−1) denote the index of the coset leader v. The index J iscoded using Slepian-Wolf codes with Y as the side information. Insteadof coding J as a whole, we code J bit-by-bit using multi-layerSlepian-Wolf coding as follows.

Assume J=(B_(m)B_(m-1) . . . B₂B₁)₂, where B_(m) is the most significantbit (MSB) of J, and B₁ is the least significant bit (LSB). A block ofthe indices may be collected. The first B₁ (i.e., a block of the firstbits from the block of indices) is encoded at rate R₁=H(B₁|Y) using aSlepian-Wolf code designed under the assumption that the correspondingdecoder has only Y as side information; then the second bit B₂ (i.e., ablock of the second bits from the block of indices) is encoded at rateR₂=H(B₂|Y,B₁) using a Slepian-Wolf code designed under the assumptionthat the corresponding decoder has only Y and B₁ as side information; .. . ; finally, the last bit B_(m) (i.e., a block of the last bits fromthe block of indices) is encoded at rate R_(m)=H(B_(m)|Y, B₁, B₂, . . ., B_(m-1)) with a Slepian-Wolf code designed under the assumption thatthe corresponding decoder has side information {Y, B₁, B₂, . . . ,B_(m-1)}. Hence the total rate of the Slepian-Wolf code isH(J|Y)=H(v|Y).

Practically, strong channel codes such as LDPC or Turbo codes areapplied as Slepian-Wolf codes. The first step in designing is todetermine the rate of the channel code to be used. Since R_(n) isequivalent to the amount of syndromes to be sent per bit, the channelcode rate is 1−R_(n). Thus the optimum rate at the n^(th) layer thatachieves Slepian-Wolf bound is 1−H(B_(n)|Y, B₁, . . . , B_(n-1)). Thismulti-layer Slepian-Wolf coding scheme is shown in FIG. 16.

As shown in FIG. 16, one embodiment of an SWC-NQ encoder includes anested lattice quantization unit 1610 and a set of Slepian-Wolf encoderSWE₁, SWE₂, . . . , SWE_(m). The nested quantization unit 1610 operateson a value of the input source X and generates the bits B₁, B₂, . . . ,B_(m-1), B_(m) of the index J as described above. The nestedquantization unit does this operation repeatedly on successive values ofthe input source, and thus, generates a stream of indices. Each of theSlepian-Wolf encoders SWE_(n), n=1, 2, . . . , m, collects a block ofthe B_(n) bits from the stream of indices and encodes this block,thereby generating an encoded block T_(n). The encoded blocks T₁, T₂, .. . , T_(m) are sent to an SWC-NQ decoder.

As shown, one embodiment of the SWC-NQ decoder includes a set ofSlepian-Wolf decoders SWD₁, SWD₂, . . . , SWD_(m) and a nestedquantization decoder 1620. Each Slepian-Wolf decoder SWD_(n), n=1, 2, .. . , m, decodes the compressed block T_(n) to recover the correspondingblock of B_(n) bits. As noted above, decoder SWD_(n) uses sideinformation {Y, B₁, B₂, . . . , B_(n-1)}. The nested quantizationdecoder 1620 operates on the blocks generated by the decoders using ablock of the Y values, as described above, to compute a block ofestimated values of the source.

C. Simulation Results.

We carry out 1-D nested lattice quantizer design for different sourceswith 10⁶ samples of X in each case. For σ_(Y) ²=1 and σ_(Z) ²=0.01, FIG.17 shows results with nested lattice quantization alone and SWC-NQ. Theformer exhibits a 3.95-9.60 dB gap from D_(WZ)(R) for R in the rangefrom 1.0 to 7.0 bits/sample (b/s), which agree with the high rate lowerbound of Theorem 1. At high rate, we observe that the gap between ourresults with ideal SWC (i.e., rate computed as H(J|Y) in the simulation)and D_(WZ)(R) is indeed 1.53 dB. With practical SWC based on irregularLDPC codes of length 10⁶ bits, this gap is 1.66-1.80 dB for R in therange from 0.93 to 5.00 b/s.

For 2-D nested lattice quantization, we use the A₂ hexagonal latticesagain with σ_(Y) ²=1 and σ_(Z) ²=0.01. FIG. 18 shows results with nestedlattice quantization alone and SWC-NQ. At high rate, the former caseexhibits a 4.06-8.48 dB gap from D_(WZ)(R) for R=1.40-5.00 b/s, again inagreement with the high rate lower bound of Theorem 1. We observe thatthe gap between our results with ideal SWC (measured in the simulation)and D_(WZ)(R) is 1.36 dB. With practical SWC based on irregular LDPCcodes (of length 10⁶ bits), this gap is 1.67-1.72 dB for R=0.95-2.45b/s.

We thus see that using optimal estimation as described herein, oursimulation results with either 1-D or 2-D nested quantization (andpractical Slepian-Wolf coding) are almost a constant gap away from theWyner-Ziv limit for a wide range of rates.

In this paper, the high-rate R-D performance of the nested latticequantization for the Wyner-Ziv coding is analyzed, with low dimensionallattice codes. The performance is away from the Wyner-Ziv bound witheach specific lattice code, and exhibits an increasing gap from theWyner-Ziv bound as the rate increases. The reason for the increase ofthe gap mainly comes from the fact that the granular component of thedistortion is an increasing function of the rate. Therefore theSlepian-Wolf coding, as a second-layer binning scheme, is applied to thequantization indices for further compression. This Slepian-Wolf codednested lattice quantization (SWC-NQ) performs at a constant gap from theWyner-Ziv bound at high rates, and the constant gap is the same as theone from ECVQ (entropy coded vector quantization) to the ideal R-Dfunction of source coding without the side information. Moreover, anon-linear estimator for the decoder is introduced, and proved to beoptimal in the sense of the MSE measurement. This non-linear estimatorhelps at low-rates, and degrades to the linear one which is assumed inthe theoretical analyses in this paper. Simulation results for 1-D and2-D cases are in agreement with the theoretical analysis.

Proof of Lower Bound (49).

Proof to establish the lower bound for the performance of quadraticGaussian SWC-NQ.

1) Rate Computation:

The rate for SWC-NQ is:

$\begin{matrix}{R = {\frac{1}{n}{{H\left( v \middle| Y \right)}.}}} & (69)\end{matrix}$

Since at high rate,

$\begin{matrix}\begin{matrix}{{P\left( {v❘Y} \right)} = {\sum\limits_{j = {- \infty}}^{\infty}{\int_{x \in {R_{j}{(v)}}}{{f_{X❘Y}(x)}\ {\mathbb{d}x}}}}} \\{= {\sum\limits_{j = {- \infty}}^{\infty}{\int_{x \in {R_{0}{(v)}}}{{f_{X❘Y}\left( {x + {c_{j}(0)}} \right)}\ {\mathbb{d}x}}}}} \\{\approx {\sum\limits_{j = {- \infty}}^{\infty}{{F_{X❘Y}\left( {v + {c_{j}(0)}} \right)}V_{1}}}} \\{{\equiv {{g(v)}V_{1}}},}\end{matrix} & (70) \\{{{{where}\mspace{14mu}{g(x)}} \equiv {\sum\limits_{j = {- \infty}}^{\infty}{f_{X❘Y}\left( {x + {c_{j}(0)}} \right)}}},{{{and}\mspace{14mu} X}❘{Y\text{∼}{{N\left( {0,\sigma_{X❘Y}^{2}} \right)}.}}}} & \;\end{matrix}$

Then the achievable rate of SWC-NQ is

$\begin{matrix}{{nR} = {{H\left( {v❘Y} \right)} = {- {\sum\limits_{v \in {\Lambda_{1}/\Lambda_{2}}}{{P\left( {v❘Y} \right)}{\log_{2}\left\lbrack {P\left( {v❘Y} \right)} \right\rbrack}}}}}} \\{\approx {- {\sum\limits_{v \in {\Lambda_{1}/\Lambda_{2}}}{{P\left( {v❘Y} \right)}{\log_{2}\left\lbrack {{g(v)}V_{1}} \right\rbrack}}}}} \\{= {{- {\sum\limits_{v \in {\Lambda_{1}/\Lambda_{2}}}{\overset{\infty}{\sum\limits_{j = {- \infty}}}{\int_{x \in {R_{0}{(v)}}}{{f_{X❘Y}\left( {x + {c_{j}(0)}} \right)}\ {\mathbb{d}x}\;\log_{2}{g(v)}}}}}} - {\log_{2}V_{1}}}} \\{\approx {{- {\sum\limits_{j = {- \infty}}^{\infty}{\sum\limits_{v \in {\Lambda_{1}/\Lambda_{2}}}{\int_{x \in {R_{0}{(v)}}}{{f_{X❘Y}\left( {x + {c_{j}(0)}} \right)}\log_{2}{g(x)}\ {\mathbb{d}x}}}}}} - {\log_{2}V_{1}}}} \\{\overset{(a)}{=}{{- {\int_{x \in R^{n}}{{f_{X❘Y}(x)}\log_{2}{g(x)}\ {\mathbb{d}x}}}} - {\log_{2}V_{1}}}}\end{matrix}$where (a) comes from the periodic property of g(·), i.e.,g(x−l)=g(x),∀l∈Λ₂. Thus the achievable rate of SWC-NQ isnR=H(v|Y)=h′(X,Λ ₂)+log₂σ_(X|Y) ^(n)−log₂ V ₁.  (71)

2) Distortion Computation: From Theorem 4.1, the average distortion ofnested lattice quantization over all realizations of (X,Y) is

$\begin{matrix}\begin{matrix}{D_{n} = {{G\left( \Lambda_{1} \right)V_{1}^{2/n}} + {\frac{1}{n}{E_{Z}\left\lbrack {{Q_{\Lambda_{2}}(z)}}^{2} \right\rbrack}}}} \\{\geq {{{G\left( \Lambda_{1} \right)}V_{1}^{2/n}} + {\frac{1}{n}{\sum\limits_{j = 1}^{\infty}\;{\left( {\left( {{2\; j} - 1} \right)\gamma_{n}V_{2}^{2/n}} \right){{u\left( \frac{j^{2}V_{2}^{2/n}\Gamma\;\left( {\frac{n}{2} + 1} \right)^{2/n}}{2\;{\pi\sigma}_{Z}^{2}} \right)}.}}}}}}\end{matrix} & (72)\end{matrix}$

Because SWC is lossless, the distortion of SWC-NQ is also D_(n).Combining D_(n) and R through V₁, we obtain the R-D performance ofSWC-NQ with a pair of n-D nested lattices (Λ₁,Λ₂) as

$\begin{matrix}{{D_{n}(R)} \geq {{{G\left( \Lambda_{1} \right)}2^{{({2/n})}{h^{\prime}{({X,\Lambda_{2}})}}}\sigma_{X|Y}^{2}2^{{- 2}\; R}} + {\frac{1}{n}{\sum\limits_{j = 1}^{\infty}\;{\left( {\left( {{2\; j} - 1} \right)\gamma_{n}V_{2}^{2/n}} \right){{u\left( \frac{j^{2}V_{2}^{2/n}\Gamma\;\left( {\frac{n}{2} + 1} \right)^{2/n}}{2\;{\pi\sigma}_{Z}^{2}} \right)}.}}}}}} & (73)\end{matrix}$

Proof of Lemma 5.1

This proof closely follows the remark 3) of [1] page 3, with some slightmodifications.

${\,_{Let}\delta} \equiv {\min\limits_{w \neq \hat{x}}\;{d\left( {w,\hat{x}} \right)}_{|Y}} > 0.$Here δ is actually the minimum of the distance between two latticepoints of Λ₂. Thus if (x,{circumflex over (x)})∈S₁,

$\begin{matrix}{\lambda \equiv {\Pr\left\{ {W \neq \hat{X}} \right\}} \leq {{E\left\lbrack {d\left( {W,\hat{X}} \right)} \right\rbrack}/\delta}\overset{(a)}{\leq}{\left( {{E\left\lbrack {d\left( {W,X} \right)} \right\rbrack} + {E\left\lbrack {d\left( {X,\hat{X}} \right)} \right\rbrack}} \right)/\delta}} & (74)\end{matrix}$where (a) comes from the triangle inequality. From Theorem 1,D=E[d(X,{circumflex over (X)})]=MSE_(g)+MSE_(ol),where MSE_(g)=E[d(W,X)] is the granular component and MSE_(ol) is theoverload component, thenλ≦2D/δ.  (75)

Now since {circumflex over (X)} is a function of V, Y, Fano's inequality[30], [31] implies that

$\begin{matrix}{{{H\left( {\left. W \middle| V \right.,Y} \right)} \leq {{{- \lambda}\;\log\;\lambda} - {\left( {1 - \lambda} \right){\log\left( {1 - \lambda} \right)}} + {\lambda\;\log\mspace{11mu}\left( {W} \right)}} \equiv {ɛ(\lambda)}},{{so}\mspace{14mu}{that}}} & (76) \\{{{H\left( V \middle| Y \right)} \geq {I\left( {W;\left. V \middle| Y \right.} \right)}} = {{{H\left( W \middle| Y \right)} - {H\left( {\left. W \middle| V \right.,Y} \right)}} \geq {{H\left( W \middle| Y \right)} - {{ɛ\left( \frac{2\; D}{\delta} \right)}.}}}} & (77)\end{matrix}$

Meanwhile, from data processing rule, we have H(V|Y)≦H(W|Y). At highrate, D→0, and

$\left. {ɛ\left( \frac{2D}{\delta} \right)}\rightarrow 0. \right.$Thus at high rate, H(V|Y)≈H(W|Y).

This claim is also verified intuitively by FIG. 13, where the slant partof each curve which corresponds to the R-D performance with a fixed V₂,or δ, approximately maintains a constant slope.

It is noted that any or all of the method embodiments described hereinmay be implemented in terms of program instructions executable by one ormore processors. The program instructions (or subsets thereof) may bestored and/or transmitted on any of various carrier media. Furthermore,the data generated by any or all of the method embodiments describedherein may be stored and/or transmitted on any of various carrier media.

Although the embodiments above have been described in considerabledetail, numerous variations and modifications will become apparent tothose skilled in the art once the above disclosure is fully appreciated.It is intended that the following claims be interpreted to embrace allsuch variations and modifications.

1. A method for generating compressed output data, the methodcomprising: a computer receiving input data from an information source;the computer applying nested lattice quantization to the input data inorder to generate intermediate data, wherein the nested latticequantization is based on a fine lattice and a coarse sublattice of thefine lattice, wherein a cell volume of the coarse sublattice minimizesan average error in estimation of the input data at a decoder given afixed nesting ratio between the coarse sublattice and the fine lattice,wherein the decoder is configured to use side information that iscorrelated with the input data to perform said estimation; and thecomputer encoding the intermediate data using an asymmetric Slepian-Wolfencoder in order to generate compressed output data representing theinput data; and storing the compressed output data.
 2. The method ofclaim 1, wherein the input data includes vectors in an n-dimensionalspace, wherein n is greater than one.
 3. The method of claim 2, whereinn equals two.
 4. The method of claim 2, wherein said applying nestedlattice quantization to the input data includes: quantizing values ofthe input data with respect to the fine lattice to determinecorresponding points of the fine lattice; and determining indices forcosets of the coarse sublattice, the cosets corresponding respectivelyto the fine lattice points, wherein the intermediate data includes saidindices.
 5. The method of claim 2, wherein the asymmetric Slepian-Wolfencoder includes a plurality of asymmetric Slepian-Wolf subencoders,wherein each of the subencoders operates on a respective bit plane ofthe intermediate data.
 6. The method of claim 5, wherein the asymmetricSlepian-Wolf subencoders operate in parallel on the respective bitplanes of the intermediate data.
 7. The method of claim 2 furthercomprising transferring the compressed output data to the decoder. 8.The method of claim 1, wherein the fine lattice is a two-dimensionalhexagonal lattice.
 9. The method of claim 1, wherein the asymmetricSlepian-Wolf encoder achieves a compression rate that approaches thelimit for Slepian-Wolf coding.
 10. The method of claim 1, wherein theasymmetric Slepian-Wolf encoder is based on one or more turbo codes. 11.A system, comprising: a quantization unit configured to receive signaldata and to perform nested lattice quantization on the signal data inorder to generate intermediate data, wherein the nested latticequantization is based on a fine lattice and a coarse sublattice of thefine lattice, wherein a cell volume of the coarse sublattice minimizesan average error in estimation of the input data at a decoder given afixed nesting ratio between the coarse sublattice and the fine lattice,wherein the decoder is configured to use side information that iscorrelated with the input data to perform said estimation; and anencoding unit coupled to the quantization unit and configured to encodethe intermediate data using a set of one or more asymmetric Slepian-Wolfencoders in order to generate compressed output data representing thesignal data.
 12. The system of claim 11, wherein the fine lattice is alattice in an n-dimensional space, wherein n is greater than one. 13.The system of claim 12, wherein the asymmetric Slepian-Wolf encoders areconfigured to operate in parallel on respective bit planes of theintermediate data.
 14. A system for generating compressed output data,the system comprising: one or more processors; and a memory that storesat least program instructions, wherein the program instructions areexecutable by the one or more processors to: perform a nested latticequantization on received input data in order to generate intermediatedata, wherein the nested lattice quantization is based on a fine latticeand a coarse sublattice of the fine lattice, wherein a cell volume ofthe coarse sublattice is selected to minimize an average error inestimation of the input data at a decoder given a fixed nesting ratiobetween the coarse sublattice and the fine lattice, wherein the decoderis configured to use side information that is correlated with the inputdata to perform said estimation; and operate on the intermediate datausing a set of one or more asymmetric Slepian-Wolf encoders in order togenerate compressed output data.
 15. The system of claim 14, wherein thefine lattice is a lattice in an n-dimensional space, wherein n isgreater than one.
 16. The system of claim 15, wherein the one or moreasymmetric Slepian-Wolf encoders include a plurality of asymmetricSlepian-Wolf encoders, wherein the program instructions are executableby the one or more processors to operate on bit planes of theintermediate data using respective ones of the encoders.
 17. The systemof claim 14, wherein at least one of the asymmetric Slepian-Wolfencoders is based on a turbo code.
 18. The system of claim 14, whereineach of the asymmetric Slepian-Wolf encoders is configured to achieve acompression rate that approaches the limit for Slepian-Wolf coding for acorresponding one of the bit planes of the intermediate data.
 19. Thesystem of claim 14, wherein the program instructions are executable bythe one or more processors to transmit the compressed output data to thedecoder.
 20. A computer-readable medium having stored thereon,computer-executable instructions that, if executed by a computer systemcause the computer system to perform a method comprising: quantizingreceived input data using nested lattice quantization in order togenerate intermediate data, wherein the nested lattice quantization isbased on a fine lattice and a coarse sublattice of the fine lattice,wherein a cell volume of the coarse sublattice minimizes an averageerror in estimation of the input data at a decoder given a fixed nestingratio between the coarse sublattice and the fine lattice, wherein thedecoder is configured to use side information that is correlated withthe input data to perform said estimation; encoding the intermediatedata using a set of one or more asymmetric Slepian-Wolf encoders inorder to generate compressed output data; and storing the compressedoutput data in a memory.
 21. The medium of claim 20, wherein the finelattice is a lattice in an n-dimensional space, wherein n is greaterthan one.
 22. The medium of claim 21, wherein the asymmetricSlepian-Wolf encoders operate in parallel on respective bit planes ofthe intermediate data.
 23. The medium of claim 21, wherein at least oneof the asymmetric Slepian-Wolf encoders is based on a turbo code. 24.The medium of claim 21, wherein the method further comprises:transferring the compressed output data to the decoder.
 25. A system,comprising: a first means for applying nested lattice quantization toreceived input data in order to generate intermediate data, wherein thenested lattice quantization is based on a fine lattice and a coarsesublattice of the fine lattice, wherein a cell volume of the coarsesublattice has been selected to minimize an average error in estimationof the input data at a decoder given a fixed nesting ratio between thecoarse sublattice and the fine lattice; and a second means for encodingthe intermediate data in order to generate compressed output data.